THE  ELEMENTS 


PRACTICAL  ASTRONOMY 


THE   ELEMENTS 


PRACTICAL   ASTRONOMY 


BY 

W.    W.    CAMPBELL 

ASTRONOMER   IN   THE    LICK   OBSERVATORY 


SECOND  EDITION,  REVISED  AND  ENLARGED 


Netofgorfe 
THE   MACMILLAN   COMPANY 

LONDON:  MACMILLAN  &  CO.,  LTD. 
1913 

All  rights  reserved 


v-      J 
Astron.  Dopt, 


COPYBIGHT,   1891, 

BY  THE  EEGISTER  PUBLISHING  CO. 

COPYRIGHT,  1899, 
BY  THE  MACMILLAN  COMPANY. 


First  published  elsewhere.     Reprinted  April,  1899;  June,  1904 ; 
November,  1906;  September,  1909  ;  September,  1910  ;  February, 


J.  S.  Cashing  &  Co.  —  Berwick  &  Smith  Co. 
Norwood,  Mass.,  U.S.A. 


PEEFACE 

TO  THE  SECOND  EDITION 

MY  experience  in  presenting  the  elements  of  practical 
astronomy  to  rather  large  classes  of  students  in  the 
University  of  Michigan  led  me  to  the  conclusion  that 
the  extensive  treatises  on  the  subject  could  not  be  used 
satisfactorily,  except  in  special  cases.  Brief  lecture  notes 
were  employed  in  preference.  Arrangements  were  made 
with  a  local  publisher  that  the  notes  should  be  written 
out  in  full  and  printed,  almost  exclusively  it  was  sup- 
posed, for  use  in  my  own  classes.  The  process  of  en- 
largement had  just  begun  when  the  call  to  my  present 
position  was  accepted.  The  completion  of  the  manu- 
script in  the  midst  of  new  and  pressing  duties  was 
extremely  difficult ;  the  text  and  the  details  of  the  treat- 
ment lacked  the  harmony  which  can  come  only  from  a 
leisurely  development  of  the  subject.  Nevertheless,  the 
first  edition  has  been  used  in  a  great  many  colleges  and 
universities  whose  astronomical  departments  are  of  the 
highest  character.  This  is  my  reason  for  carefully  revis- 
ing and  slightly  enlarging  the  book  for  a  second  edition. 

A  word  concerning  the  limitations  of  the  book  may 
not  be  out  of  place.  The  field  of  practical  astronomy 
has  become  very  extensive,  embracing  essentially  all  the 
work  carried  on  in  our  astronomical  observatories.  It 


459801 


VI  PREFACE 

includes  the  photographic  charting  of  the  stars,  the  spec- 
troscopic  determination  of  stellar  motions,  the  determi- 
nation of  solar  parallax  from  heliometer  observations  of 
the  asteroids,  the  construction  of  empirical  formulae  and 
tables  for  computing  atmospheric  refraction,  and  scores 
of  other  operations  of  equally  high  character.  These, 
however,  can  best  be  described  as  special  problems,  re- 
quiring prolonged  efforts  on  the  part  of  professional 
astronomers ;  in  fact,  the  solution  of  a  single  problem 
often  severely  taxes  the  combined  resources  of  a  number 
of  leading  observatories.  While  it  is  evident  that  a 
discussion  of  the  methods  employed  in  solving  special 
problems  must  be  looked  for  in  special  treatises  and  in 
the  journals,  yet  these  methods  are  all  developed  from 
the  elements  of  astronomy,  of  physics,  and  of  the  other 
related  sciences.  It  is  intended  that  this  book  shall 
contain  the  elements  of  practical  astronomy,  with  numer- 
ous applications  to  the  problems  first  requiring  solution. 
It  is  believed  that  the  methods  of  observing  employed 
are  illustrations  of  the  best  modern  practice.  The 
methods  of  reduction  are  intended  to  be  exact  to  the 
extent  that  none  of  the  value  and  precision  of  the  obser- 
vations will  be  sacrificed  in  the  computations ;  further 
refinement  would  be  superfluous,  and  misleading  to  the 
inexperienced.  The  demonstrations  are  direct  and  fun- 
damental, except  in  the  case  of  refraction.  I'he  scien- 
tific basis  of  the  subject  of  refraction  is  largely  physical, 
and  the  astronomical  superstructure  is  almost  wholly 
empirical.  For  these  reasons,  the  proper  proportions  of 
the  subject  with  reference  to  the  rest  of  the  book  seem 
to  be  preserved  by  the  insertion  of  the  final  formulae. 


PREFACE  Vll 

An  attempt  has  been  made  to  give  credit  for  methods 
which  have  not  yet  found  their  way  into  general  practice. 

The  illustrations  of  modern  instruments  are  from  cuts 
kindly  furnished  by  the  makers,  viz.:  those  for  Fig- 
ures 15,  20,  21  and  24  by  Fauth  &  Co.,  Washington  ; 
those  for  Figures  14,  17  and  26  by  C.  L.  Berger  & 
Sons,  Boston;  that  for  Figure  10  by  the  Keuffel  & 
Esser  Co.,  New  York  ;  and  that  for  Figure  27  by  War- 
ner &  Swasey,  Cleveland.  Figure  25,  of  a  Repsold 
meridian  circle,  is  copied  with  permission  from  Baron 
A.  v.  Schweiger-Lerchenfeld's  Atlas  der  Himmelskunde 
(Vienna). 

The  author  is  indebted  to  his  colleagues,  Professors 
Schaeberle,  Tucker,  Hussey  and  Perrine,  for  valuable 
suggestions  and  assistance. 

W.  W.  CAMPBELL. 

LICK  OBSERVATORY, 

UNIVERSITY  OF  CALIFORNIA, 

January,  1899. 


CONTENTS 


CHAPTER  I 

PAGK 

THE  CELESTIAL  SPHERE 1 

Definitions 2 

Systems  of  coordinates .7 

Transformation  of  coordinates 8 

Distance  between  two  stars  ...  14 


CHAPTER  II 

TIME  —  Definitions 15 

Conversion  of  time 18 


CHAPTER  III 

CORRECTION  OF  OBSERVATIONS  —  Form  and  dimensions  of  the 

earth    ....   - 23 

Parallax  —  Definitions 25 

In  zenith  distance  .........  26 

In  azimuth  and  zenith  distance 27 

In  right  ascension  and  declination 31 

Refraction 32 

General  laws — In  zenith  distance 33 

In  right  ascension  and  declination  .....  36 

Dip  of  the  horizon          .        .         .                 36 

Semidiameter  —  Of  the  moon r  .  38 

Contraction  of  semidiameters  by  refraction  ....  39 

Aberration 40 

Diurnal  aberration  in  hour  angle  and  declination  .  .  41 

Diurnal  aberration  in  azimuth  and  altitude  ....  43 

Sequence  and  degree  of  corrections  .  .  .  .  -  .  .43 

ix 


CONTENTS 


CHAPTER  IV 

PAGE 

PRECESSION  —  NUTATION  —  ANNUAL    ABERRATION  —  PROPER 

MOTION 44 

Star  places 45 

Precession       ...........  46 

Annual  precession  .........  50 

Proper  motion        .         . 54 

Determination  by  the  method  of  least  squares      ...  58 

Reduction  to  apparent  place 61 

CHAPTER  V 

ANGLE  AND  TIME  MEASUREMENT  — The  vernier  ...  65 

The  reading  microscope 67 

Eccentricity 69 

The  micrometer  .  . 70 

Determination  of  the  value  of  a  revolution  ....  72 

The  level 79 

Determination  of  the  value  of  a  division  .  .  .  .81 

The  chronometer ,83 

Eye  and  ear  method  of  observing .         .         .         ....  85 

The  astronomical  clock  —  The  chronograph  ....  86 


CHAPTER  VI 

THE  SEXTANT — Description 89 

General  principles  of  the  sextant  .        .        .        .        .        .         .91 

Methods  of  observing  with  the  sextant 92 

Adjustments  of  the  sextant 94 

Corrections  to  sextant  readings 96 

Determination  of  time 101 

By  equal  altitudes  of  a  fixed  star 101 

By  equal  altitudes  of  the  sun 102 

By  a  single  altitude  of  a  star 106 

By  a  single  altitude  of  the  sun 107 

Determination  of  geographical  latitude 109 

By  a  meridian  altitude  of  a  star  or  the  sun  ....     109 

By  an  altitude  of  a  star 110 

By  circummeridian  altitudes  .......     112 

Determination  of  geographical  longitude      .        .        .        .        .115 

By  lunar  distances  .         .         .        .        .        .        .        .        .115 


CONTENTS  XI 
CHAPTER  VH 

PAGK 

THE  TRANSIT  INSTRUMENT  —  Description 122 

Definition  of  instrumental  constants 127 

General  equations  ..........  129 

Determination  of  the  wire  intervals      .        .        .        .        .        .  132 

Determination  of  the  level  constant ,      .  134 

Determination  of  the  collimation  constant 137 

Determination  of  the  azimuth  constant        .....  142 

Meridian  mark,  or  mire 143 

Adjustments -.  144 

Determination  of  time 146 

Reduction  by  the  method  of  least  squares 152 

Correction  for  flexure     .        .         .         .         .        .        .        .        .157 

Personal  equation  .         .         .         .  .        .        .        .         .157 

Determination  of  geographical  longitude 159 

By  transportation  of  chronometers 159 

By  the  electric  telegraph 160 

By  the  heliotrope    .        .        ...        .        .        .        .        .  163 

By  moon  culminations 164 

CHAPTER  VIII 

THE  ZENITH  TELESCOPE 167 

Determination  of  geographical  latitude  by  Talcott's  method       .  167 

Combination  of  results  by  the  method  of  least  squares        .        .  173 


CHAPTER  IX 

THE  MERIDIAN  CIRCLE  —  Description 175 

Determination  of  right  ascension 177 

Determination  of  flexure 181 

Errors  of  graduation 183 

Determination  of  declination  .  187 


CHAPTER  X 

ASTRONOMICAL  AZIMUTH •  .        •  191 

Azimuth  by  a  circumpolar  star  near  elongation  ....  193 

Azimuth,  by  Polaris  observed  at  any  hour  angle  ....  199 


Xll  CONTENTS 


CHAPTER  XI 

PAGE 

THE  SURVEYOR'S  TRANSIT 203 

Determination  of  time 203 

By  equal  altitudes  of  a  star 203 

By  a  single  altitude  of  a  star  .......  205 

By  a  single  altitude  of  the  sun 207 

Determination  of  geographical  latitude 20!) 

By  a  meridian  altitude  of  a  star 209 

By  a  meridian  altitude  of  the  sun 210 

Determination  of  azimuth 211 


CHAPTER  XII 

THE  EQUATORIAL  —  Description 212 

Adjustments 214 

Magnifying  power .  220 

Field  of  view 221 

Determination  of  apparent  place  of  an  object       ....  223 

By  the  method  of  micrometer  transits  .         .        .   •     .         .  223 

By  the  method  of  direct  micrometer  measurement        .        .  229 

Determination  of  position  angle  and  distance       ....  233 

The  ring  micrometer .        .        .  236- 


APPENDICES 

A.  Hints  on  computing 241 

B.  Interpolation  formulae    ........  244- 

C.  Combination  and  comparison  of  observations        .        .        .  247 

D.  Objects  for  the  telescope 251 

TABLE      I.     Pulkowa  refraction  tables 254 

TABLE    II.     Pulkowa  mean  refractions 257 

TABLE  III.     Reduction  to  the  meridian,  or  to  elongation  .        .  25& 


INDEX .  261 


PRACTICAL  ASTRONOMY 


CHAPTER   I 

DEFINITIONS  — SYSTEMS  OF   COORDINATES— TRANS- 
FORMATION  OF   COORDINATES 

1.  The  heavenly  bodies  appear  to  us  as  if  they  were 
situated  on  the  surface  of  a  sphere  of  indefinitely  great 
radius,  whose  center  is  at  the  point  of  observation.  Their 
directions  from  us  are  constantly  changing.  They  all 
appear  to  move  from  east  to  west  at  such  a  rate  as  to 
make  one  complete  revolution  in  about  twenty-four  hours. 
This  is  due  to  the  diurnal  rotation  of  the  earth.  The  sun 
appears  to  move  eastward  among  the  stars  at  such  a  rate 
as  to  make  one  revolution  per  year.  This  is  caused  by 
the  annual  revolution  of  the  earth  around  the  sun.  The 
moon  and  the  various  planets  have  motions  characteristic 
of  the  orbits  which  they  describe.  Measurements  with 
instruments  of  precision  enable  us  to  detect  other  motions 
which,  we  shall  see  later,  are  conveniently  divided  into 
two  classes :  those  due  to  parallax,  refraction,  and  diurnal 
aberration,  which  depend  upon  the  observer's  geographi- 
cal position;  and  those  due  to  precession,  nutation,  annual 
aberration,  and  proper  motion,  which  are  independent  of 
the  observer's  position. 

From  data  furnished  by  systematic  observations  it  has 
been  shown  that  these  motions  occur  in  accordance  with 
well-defined  physical  laws.  It  is  therefore  possible  to 
compute  the  position  of  a  celestial  object  for  any  given 


2  PRACTICAL   ASTRONOMY 

instant.  A  table  giving  at  equal  intervals  of  time  the 
places  of  a  body  as  affected  by  the  second  class  of  motions 
mentioned  above,  is  called  an  ephemeris  of  the  body.  The 
astronomical  annuals*  furnish  accurate  ephemerides  of 
the  principal  celestial  objects '  several  years  in  advance. 
If  an  observer  knows  his  position  on  the  earth,  he  can, 
from  data  furnished  by  the  ephemerides,  compute  the 
direction  of  a  starf  at  any  instant.  Conversely,  by 
observing  the  directions  of  the  •  stars  with  suitable  instru- 
ments, he  can  determine  the  time  and  his  geographical 
position.  It  is  with  this  converse  problem  that  we  are 
principally  concerned. 

DEFINITIONS 

2.  The  sphere  on  whose  surface  the  stars  appear  to  be 
situated  is  called  the  celestial  sphere.  Any  plane  passing 
through  the  point  of  observation  cuts  the  celestial  sphere 
in  a  great  circle.  Since  the  radius  is  indefinitely  great, 
all  parallel  planes  whose  distances  apart  are  finite  cut  the 
sphere  in  the  same  great  circle. 

In  order  to  determine  the  position  of  a  point  on  the 
sphere  and  express  the  relation  existing  between  two  or 
more  points,  the  circles,  lines,  points  and  terms  defined 
below  are  in  current  use. 

The  horizon  is  the  great  circle  of  the  sphere  whose  plane 
passes  through  the  point  of  observation  and  is  perpen- 
dicular to  the  plumb-line. 

The  produced  plumb-line,  or  vertical  line,  cuts  the 
sphere  above  in  the  zenith  and  below  in  the  nadir.  The 

*  The  principal  annuals  are  the  American  Ephemeris  and  Nautical 
Almanac,  the  Berliner  Astronomisches  Jalirbuch,  the  (British)  Nautical 
Almanac,  and  the  Connaissance  des  Temps.  Unless  otherwise  specified 
we  shall  refer  to  the  first  of  these,  and  call  it  the  American  Ephemeris, 
or  the  Ephemeris. 

t  For  convenience  we  shall  use  star,  point  or  body  to  denote  any 
celestial  object. 


DEFINITIONS  O 

zenith  and  nadir  are  the  poles  of  the  horizon,  and  all  great 
circles  passing  through  them  are  called  vertical  circles. 

The  points  of  the  horizon  directly  south,  west,  north 
and  east  of  the  observer  are  called,  respectively,  the  south, 
west,  north  and  east  points. 

The  meridian  is  the  vertical  circle  which  passes  through 
the  south  and  north  points. 

The  prime  vertical  is  the  vertical  circle  which  passes 
through  the  east  and  west  points. 

The  altitude  of  a  point  is  its  distance  from  the  horizon, 
measured  on  the  vertical  circle  passing  through  the  point. 
Distances  above  the  horizon  are  4- ;  below,  — .  The  alti- 
tudes of  all  points  on  the  sphere  are  included  between 
0°  and  +  90°,  and  0°  and  -  90°.  Instead  of  the  alti- 
tude, it  is  frequently  convenient  to  use  the  zenith  distance, 
which  is  the  distance  of  the  point  from  the  zenith,  meas- 
ured on  the  vertical  circle  of  the  point.  It  is  the  com- 
plement of  the  altitude.  The  zenith  distances  of  all 
points  on  the  sphere  lie  between  0°  and  + 180°. 

The  azimuth  of  a  point  is  the  arc  of  the  horizon  inter- 
cepted between  the  vertical  circle  of  the  point  and  some 
fixed  point  assumed  as  origin.  With  astronomers  it  is 
customary  to  reckon  azimuth  from  the  south  point  around 
to  the  west  through  360°.  Surveyors  frequently  reckon 
from  the  north  point. 

The  celestial  equator  is  the  great  circle  of  the  sphere 
whose  plane  is  perpendicular  to  the  earth's  axis.  It 
therefore  coincides  with  or  is  parallel  to  the  terrestrial 
equator. 

The  earth's  axis  produced  is  the  axis  of  the  celestial 
sphere.  It  cuts  the  sphere  in  the  north  and  south  poles 
of  the  equator.  We  shall  for  brevity  call  them  the  north 
and  south  poles. 

All  great  circles  passing  through  the  north  and  south 
poles  are  called  hour  circles.  The  hour  circle  passing 
through  the  zenith  coincides  with  the  meridian. 


4  PRACTICAL   ASTRONOMY 

The  declination  of  a  point  is  its  distance  from  the 
equator,  measured  on  the  hour  circle  passing  through  the 
point.  Distances  north  are  +  ;  south,  — .  The  declina- 
tions of  all  points  on  the  sphere  are  included  between 
0°  and  +  90°,  and  0°  and  -  90°. 

Instead  of  the  declination,  it  is  sometimes  convenient 
to  use  the  north  polar  distance,  which  is  the  distance  of 
a  point  from  the  north  pole,  measured  on  the  hour  circle 
of  the  point.  It  is  therefore  the  complement  of  the 
declination-  The  north  polar  distances  of  all  points  lie 
between  0°  and  +  180°. 

The  hour  angle  of  a  point  is  the  arc  of  the  equator 
intercepted  between  the  meridian,  or  south  point  of  the 
equator,  and  the  hour  circle  passing  through  the  point. 
In  practice,  however,  it  is  customary  to  consider  the  hour 
angle  as  the  equivalent  angle  at  the  north  pole  between 
the  meridian  and  hour  circle.  It  is  reckoned  from  the 
meridian  around  to  the  west  through  24  hours,  or  360°. 

The  ecliptic  is  the  great  circle  of  the  sphere  formed  by 
the  plane  of  the  earth's  orbit ;  or,  it  is  the  great  circle 
described  by  the  apparent  annual  motion  of  the  sun.  It 
intersects  the  equator  in  two  points  called  the  equinoxes. 

The  vernal  equinox  is  that  point  through  which  the 
sun  appears  to  pass  in  going  from  the  south  to  the  north 
side  of  the  equator  (about  March  20). 

The  autumnal  equinox  is  that  point  through  which  the 
sun  appears  to  pass  in  going  from  the  north  to  the  south 
side  of  the  equator  (about  Sept.  22). 

The  solstices  are  the  points  of  the  ecliptic  90°  from 
the  equinoxes.  The  sun  is  in  the  summer  solstice  about 
June  21  ;  in  the  winter  solstice  about  Dec.  21. 

The  equinoctial  colure  is  the  hour  circle  passing  through 
the  equinoxes.  The  solstitial  colure  is  the  hour  circle 
passing  through  the  solstices. 

The  angle  between  the  equator  and  ecliptic  is  called  the 
obliquity  of  the  ecliptic. 


DEFINITIONS  5 

The  right  ascension  of  a  point  is  the  arc  of  the  celestial 
equator  intercepted  between  the  vernal  equinox  and  the 
hour  circle  of  the  point.  It  is  measured  from  the  vernal 
equinox  toward  the  east  through  24  hours,  or  360°. 

The  sidereal  time  at  any  point  of  observation  is  equal  to 
the  right  ascension  of  the  observer's  meridian.  It  is  like- 
wise equal  to  the  hour  angle  of  the  vernal  equinox. 

Great  circles  perpendicular  to  the  ecliptic  are  called 
latitude  circles. 

The  latitude  of  a  point  is  its  distance  from  the  ecliptic, 
measured  on  the  latitude  circle  passing  through  the  point. 
Distances  north  are  -f- ;  south,  — .  The  latitudes  of  all 
points  on  the  sphere  are  included  between  0°  and  +  90°, 
and  0°  and  -  90°. 

The  longitude  of  a  point  is  the  arc  of  the  ecliptic  inter- 
cepted between  the  vernal  equinox  and  the  latitude  circle 
of  the  point.  It  is  measured  from  the  vernal  equinox 
toward  the  east  through  360°.  • 

The  position  of  an  observer  on  the  earth's  surface  is 
defined  by  his  geographical  latitude  and  longitude. 

The  geographical  latitude  of  a  place  is  the  declination  of 
the  zenith  of  the  place.  It  is  also  equal  to  the  altitude  of 
the  north  pole.  Latitudes  of  places  north  of  the  equator 
are  -f ;  south,  — . 

The  geographical  longitude  of  a  place  is  the  arc  of  the 
equator  intercepted  between  the  meridian  of  the  place  and 
the  meridian  of  some  other  place  assumed  as  origin.  It  is 
customary  to  reckon  longitudes  west  (+)  and  east  (  — ) 
from  the  meridian  of  Greenwich,  through  12  hours,  or 
180°. 

The  preceding  definitions  are  illustrated  by  Fig.  1. 
The  celestial  sphere  is  orthogonally  projected  on  the 
plane  of  the  horizon,  SWNE.  The  zenith  Z  is  projected 
on  the  point  of  observation.  NZ8  is  the  meridian  ;  EZW 
the  prime  vertical;  WVQE  the  equator;  VLBV  the 
ecliptic ;  P  the  north  pole ;  P1  the  north  pole  of  the 


6 


PRACTICAL  ASTRONOMY 


ecliptic ;  V  the  vernal  equinox ;  V  the  autumnal  equi- 
nox ;  VP  the  equinoctial  colure ;  CPP'  the  solstitial 
colure;  BC=PP'  =  .SF(7*  =  the  obliquity  of  the  ecliptic. 
Let  0  be  any  point  on  the  sphere  ;  then  ZOA  is  its 
vertical  circle ;  MOP  its  hour  circle  ;  L  OP'  its  latitude 
circle.  The  position  of  the  point  0  is  defined  by  the 
following  arcs,  called  spherical  coordinates : 


A0  =  Altitude,  h, 

ZO  =  Zenith  distance,  2, 
SA  =  SZA  =  Azimuth,  A, 

MO  =  Declination,  8, 

PO  =  North  polar  distance,  P, 
QM  =  QPM  =  Hour  angle,  t, 

VM  =  Right  ascension,  a, 
VQ  =  VPQ  =  Sidereal  time,  0, 

LO  =  Latitude,  /3, 

VL  =  Longitude,  X, 
PP'  =  BVC  =  Obliquity  of  the  ecliptic,  e. 


*  To  be  exact,  we  should  say  that  the  angle  BVC  is  measured  by  the 
arc  EC  and  by  the  arc  PP' ;  but  in  the  operations  of  practical  astronomy 
the  distinction  is  seldom  made. 


SYSTEMS    OF    COORDINATES 


3.  It  will  be  observed  that  the  horizon,  equator  and 
ecliptic  are  of  fundamental  importance.  They  are  called 
primary  circles.  Vertical  circles,  hour  circles  and  lati- 
tude circles,  which  are  respectively  perpendicular  to  them, 
are  called  secondary  circles.  Two  spherical  coordinates, 
one  measured  on  a  primary  circle,  the  other  on  its  sec- 
ondary, are  necessary  and  sufficient  to  determine  com- 
pletely the  direction  of  a  point ;  and  from  the  definitions 
just  given,  to  meet  the  requirements  of  astronomical  work, 
we  formulate  four 

SYSTEMS   OF   COORDINATES 


CIRCLES  OP  REFERENCE 

COORDINATES 

System 

Primary 

Secondary 

Primary 

Secondary 

I 

Horizon 

Vertical  circle 

Azimuth 

Altitude 

II 
III 
IV 

Equator 
Equator 
Ecliptic 

Hour  circle 
Hour  circle 
Latitude  circle 

Hour  angle 
Right  ascension 
Longitude 

Declination 
Declination 
Latitude 

The  altitude,  azimuth  and  hour  angle  of  a  star  are  con- 
tinually changing.  They  are  functions  of  the  time  and 
the  observer's  position.  Hence  they  are  adapted  to  the 
determinations  of  time,  azimuth  and  geographical  latitude 
and  longitude.  Right  ascension  and  declination  are  nearly 
Independent  of  the  observer's  position,  and  vary  with  the 
time.  They  are  largely  used  for  recording  the  relative 
positions  of  stars,  and  in  ephemerides.  Latitude  and 
longitude  are  also  nearly  independent  of  the  observer's 
position,  but  are  employed  almost  exclusively  in  theoreti- 
cal astronomy. 

In  the  solution  of  many  problems  of  practical  astron- 
omy, it  is  required  that  the  coordinates  of  a  point  in  one 
system  be  transformed  into  the  corresponding  coordinates 
in  another  system. 


8  PRACTICAL  ASTRONOMY 

TRANSFORMATION  OF   COORDINATES 

4.  Given  the  altitude  and  azimuth  of  a  star,  required  its 
declination  and  hour  angle. 

This  transformation  is  effected  by  solving  the  spherical 
triangle  PZO,  Fig.  1,  whose  vertices  are  at  the  pole,  the 
star,  and  the  zenith.  Three  parts  of  this  triangle  are 
known  ;  ZO  the  zenith  distance  or  complement  of  the 
given  altitude,  PZO  the  supplement  of  the  given  azimuth, 
and  PZ  the  complement  of  the  given  latitude;  from  which, 
by  the  methods  of  Spherical  Trigonometry,  we  can  find 
P  0  the  complement  of  the  required  declination,  and  ZPO 
the  required  hour  angle. 

For  any  spherical  triangle  ABC  we  have  \_0hauvenet' s 
Sph.  Trig.,  §  114]  the  general  equations 

pos  a  =  cos  b  cos  c  +  sin  b  sin  c  cos  A,  (1) 

sin  a  cos  B  =  cos  b  sin  c  —  sin  b  cos  c  cos  A,  (2) 

sin  a  sin  B  =  sin  b  sin  A.  <  (3) 

To  adapt  these  equations  to  the  triangle  POZ,  let 

A  =  PZO  =  180°  -  A,       a  =  PO     =  9.0°  -  8, 
b  =  ZO     =    90°  -  h,         B  =  ZPO  -  t, 
c   =PZ     =    90°-<£. 

Then  (1),  (2)  and  (3)  become 

sin  8  =  sin  h  sin  </>  —  cos  h  cos  <j>  cos  A,  (4) 

cos  8  cos  t  =  sin  h  cos  <f>  -f  cos  h  sin  <j>  cos  At  (5) 

cos  8  sin  t  =  cos  h  sin  A,  (6) 

which  enable  us  to  find  S  and  t. 

If    h    be    replaced   by   its   equivalent,    90°  —  z,   these 

become 

sin  8  =  cos  z  sin  <f>  —  sin  z  cos  <j>  cos  A,  (7) 

cos 8  cos  t  =  cos  z  cos  <f>  +  sin  z  sin  <f>  cos  A,  (8) 

cos  8  sin  t  —  sin  z  sin  A .  (9) 

These  equations  are  not  adapted  to  logarithmic  compu- 
tations (unless  addition  and  subtraction  logarithmic  tables 
are  employed),  and  they  will  be  further  transformed. 


TRANSFORMATION    OF   COORDINATES 

Let  m  be  a  positive  abstract  quantity,  and  M  an  angle 

such  that 

m  sin  M  =  sin  z  cos  A,  (10) 

m  cosM  =  cosz,  (11) 

which  conditions  may  always  be  satisfied  [Chauvenef s 
Plane  Trig.,  §  174],  Substituting  these  in  (7),  (8)  and 
(9),  they  become 

sin  8  =  m  sin  (<£  —  -M), 
cos  8  cos  t  —  m  cos  (<£  —  M), 
cos 8  sin  t  =  sin  zsinA. 

From  these  and  (10)  and  (11)  there  result 

tan  M  =  tan  z  cos  A ,  (12) 

tan,     =  ta"lSin,fv  (13) 

cos  (<£  -  M) 

tan  8    =  tan  (<£  -  M )  cos  *,  (14) 

which  completely  effect  the  transformation.  The  com- 
putations are  partially  checked  by  (9). 

The  quadrant  of  M  is  determined  by  (10)  and  (11). 
t  is  greater  or  less  than  180°  according  as  A  is  greater  or 
less  than  180°,  since  both  terminate  on  the  same  side  of 
the  meridian.  The  quadrant  of  8  is  fixed  by  (14). 

Example.  At  Ann  Arbor,  1891  March  13  the  altitude 
of  Regulus  is  +32°  10'  15". 4,  and  the  azimuth  is  283° 
5'  6". 4.  Find  the  declination  and  hour  angle.  [For 
instructions  in  the  art  of  computing,  see  Appendix  A.] 


<t> 

+    42°  16'  48"  .0 

(Amer.  Ephem., 

p.  482) 

z 

57  49  44  .6 

tan  (<£  -  M) 

9.616907 

A 

283     5     6  .4 

cos* 

9.728805 

tanz 

0.201332 

8 

+  12°29'56".7 

cos  A 

9.354873 

M 

19°  47'  41".2 

Proof 

<£ 

42   16  48  .0 

sin  z 

9.927608 

tan^l 

0.633702n 

sin  A 

9.988575n 

sin  M 

9.529754 

cosec  t 

0.073401n 

sec  (<£  -M) 

0.034338 

sec  8 

0.010416 

taut 

0.197794n 

logl 

0.000000 

t 

302°  22'  54".0 

t 

20*    9™  31s  .6 

10  PRACTICAL   ASTRONOMY 

5.    Given  the  declination  and  hour  angle  of  a  star,  required 
its  azimuth  and  zenith  distance. 

In  the  general  equations  (1),  (2)  and  (3)  let 

b  =   90°  -  8,     c  =  90°  -  <£,    A  =  t, 


and  they  become 

cos  z  =      sin  8  sin  <£  -f  cos  8  cos  <f>  cos  f, 
sin  z  cos  A  =  —  sin  8  cos  <f>  +  cos  8  sin  <£  cos  t,  (16) 

sin  z  sin  A  =      cos  8  sin  t.  ,  (17) 

To  transform  them  for  logarithmic  computation,  put 

n  sin  N  =  sin  8,  (18) 

n  cos  N  =  cos  8  cos  t.  (19) 

Whence 

tan^=^  (20) 

cos* 

ta"'COS^,  (21) 

sin  (<£  -  N) 


tan  z   =^LL^ — ±^.,  (22) 

cos  A 

which  effect  the  transformation.     (17)  furnishes  a  partial 
check  on  the  computations. 

Example.  At  Ann  Arbor,  1891  March  13,  when  the 
hour  angle  of  Regulus  is  20A  9m  31*.  6,  what  are  the  azimuth 
and  zenith  distance  ? 

8  +  12°  29'  56".4  (Amer.  Ephem.,  p.  332) 

t  302  22  54  .0 

tan  8  9.345719  tan  (<£  -  N)  9.556204 

cos*  9.728805  cos  A  9.354873 

N  22°  29'  7".9  z  57°49'44".6 

<t>  42  16  48  .0  '  Proof 

tan*  0.197794n  cos  8  9.989583 

cos  AT  9.965661  sin*  9.926599n 

cosec  (<£  -  N)  0.470247  cosec  A  0.011425n 

tan  A  0.633702n  cosec  z  0.072392 

A  283°5'6".4  log  1  9.999999 


TRANSFORMATION   OF   COORDINATES  11 

6.  The  angle  POZ,  Fig.  1,  between  the  hour  and  vertical 
circles  of  a  star,  is  called  the  star's  parallactic  angle.  Let 
q  represent  it. 

To  find  the  parallactic  angle  when  2,  A,  and  <£  are  given, 
we  have,  from  (1),  (2)  and  (3), 

sin  8  =  cos  z  sin  <j>  —  sin  z  cos  <f>  cos  A,  (23) 

cos  8  cos  q  =  sin  z  sin  </>  +  cos  z  cos  <j!>  cos  A,  (24) 

cos  8  sin  q  —  sin  A  cos  <f>.  (25) 

Assume 

k  sin  K  =  sin  <£,  (26) 

k  cos  K  =  cos  <f>  cos  A,  (27) 

and  we  obtain 

tan  tf  =  *2!Li  (28) 

cos  ^4 

tang  =  tan^COB^.  (29) 

cos  (/£  -  z) 

The  quadrant  of  #  is  determined  by  (25)  and  (29). 

To  find  the  parallactic  angle  and  zenith  distance  when 
8,  £,  and  <£  are  given,  we  have,  from  (1),  (2)  and  (3), 

cos  z  =  sin  8  sin  <f>  +  cos  8  cos  <£  cos  t,  (30) 

sin  z  cos  q  =  cos  8  sin  <f>  —  sin  8  cos  <f>  cos  t,  (31) 

sin  z  sin  <?  =  sin  t  cos  <£.  (32) 

Assume 

/  sin  L  =  cos  <ft  cos  *,  (33) 

I  cos  i  =  sin  <£,  (34) 

and  we  obtain 

tan  L  —  cot  <£  cos  t,  (35) 

tang  =tantsinZ  (36) 

cos(S+£) 

tanz  =cot(S  +  £).  (37) 

The  computations  may  be  partially  checked  by  (32). 

The  values  of  q  obtained  from  the  data  of  §  4  and  §  5 
are  equal  to  each  other,  and  to  312°  25'  33".5. 


\ 

12  PRACTICAL    ASTRONOMY 

7.    Given  the  declination  and  zenith  distance  of  a  star, 
required  its  hour  angle. 

If  a,  b  and  c  are  the  sides  and  A  an  angle  of  a  spherical 
triangle,  we  have  [  Chauvenefs  Sph.  Trig.,  §  18] 


tan  *  A  =  ±    /Bin(B-b)8iu(8-c) 
\       sin  s  sin  (s  —  a) 

in  which  s  =  J(a  H-  b  -f  c).     If  in  this  we  substitute  from 
triangle  POZ 

A  =  t,    a  =  2,     b  =  90°-8,    c  =  90°  -  <£, 
it  reduces  to 


tan  It  =  ±      gLJ      +  (*  -  *)]  si"  K*-(*-  *)].         (38) 
Vec*i[fl  +  (<£  +  8)]  cosi[z  -  (<£  +  8)] 

Similarly,  it  can  be  shown  that 


sin i I  =  ±    /si»  *[«+(»-  *)]  ^" K«  -  (»  -^  (39) 

\  COS  <^>  COS  8 

To  determine  the  quadrant  of  t  it  must  appear  from  the 
data  of  the  problem  whether  the  star  is  west  or  east  of  the 
meridian.  If  it  is  west,  ^t  is  in  the  first  quadrant ;  if  east, 
^t  is  in  the  second.  Applications  of  formula  (38)  may  be 
found  in  §§  81  and  82. 

8.  Criven  the  hour  angle  of  a  star,  required  its  right 
ascension,  and  vice  versa;  the  sidereal  time  in  both  cases 
being  known.  . 

In  Fig.  1,  for  any  star  0  we  have 

VM  —  right  ascension  of  star  =  a, 
MQ  =  hour  angle  of  star  =  t, 

VQ  =  sidereal  time  =  0. 
Then 

a  =  0-t,  (40) 

and 

t  =  0  -  a,  (41) 

which  effect  the  transformations. 


TRANSFORMATION    OF   COORDINATES  13 

Applications  of  (40)  and  (41)  are  numerous  throughout 
the  book. 

9.  Given  the  right  ascension  and  declination  of  a  star, 
required  its  longitude  and  latitude,  and  vice  versa. 

The  transformation  formulae  are  obtained  by  applying 
the  general  equations  (1),  (2)  and  (3)  to  the  triangle 
POP',  Fig.  1,  in  which 

OP  =  90°  -  8,      OP'  =  90°  -  /?,     GPP'  =  90°  +  a, 
OP'P  =  90°  -  A,     PPf  -  obliquity  of  ecliptic  =  c. 

In  order  to  adapt  the  resulting  equations  to  logarithmic 
computation,  assume 

/sinF  =  sin8,  (42) 

fcosF=  cos  8  sin  a,  (43) 

and  we  shall  obtain 

tanF  =  *5!L?,  (44) 

sin  a 

na,  (45) 


tan  J3  =  tan  (F  -  e)  sin  X.  (46) 

The   computations   may  be   partially  checked  by  the 

equation 

cos  8  sin  a  sec  F  cos  (F  —  e)  cosec  X  sec  (3  —  1,  (47) 

which  is  derived  without  difficulty  from  the  transforma- 
tion formulae. 

Example.    The  coordinates  of  Regulus  on  1891  March 

13  are 

a  =  150°  38'  43".5,     8  =  +  12°  29'  56"  .4. 

What  are  the  corresponding  longitude  and  latitude  ? 

The  necessary  value  of  e,  furnished  by  the  American 
Ephemeris,  page  278,  is  23°  27'  16".  0.  The  resulting 
coordinates  are 

X  =  148°  19'  3".l,    /J  =  +  0°27'40".5. 


14  PRACTICAL   ASTRONOMY 

10.  Griven  the  right  ascensions  and  declinations  of  two 
stars,  required  the  distance  between  them. 

Let  the  coordinates  of  the  stars  be  a',  £',  and  a",  S", 
and  d  the  required  distance.  In  the  spherical  triangle 
whose  vertices  are  at  the  two  stars  and  the  pole,  the  sides 
are  90°  —  Sf,  90°  —  Bff  and  d,  and  the  angle  at  the  pole  is 
arf  —  af.  Let  B'  represent  the  angle  opposite  90°  —  8'. 
If  in  (1),  (2)  and  (3)  we  put 

a  =  rf,  B  =  £',   b  =  90°-S',   c  =  90°-8",   A  =  a."  -  a', 

they  become 

cos  d  =  sin  B'  sin  8"  +  cos  8'  cos  8"  cos  (a"  -  a'),  (48) 

sin  d  cos  B'  =  sin  8'  cos  8"  -  cos  8'  sin  8"  cos  (a"  -  a'),  (49) 

sin  d  sin  B'  =  cos  8'  sin  (a"  —  a').  (50) 

If  d  can  be-  determined  from  its  cosine  with  sufficient 
precision,  (48)  will  give  the  required  distance;  otherwise 
it  should  be  determined  from  the  tangent.  If  we  assume 

g  sin  G  =  cos  8'  cos  (a"  —  a'),  "  (51) 

g  cos  G  =  sin  8',  (52) 
we  shall  find  that 

tan  G  =  cot  8'  cos  (a"  -  a'),  (53) 


tand=co  .  (55) 

cos  B' 

(50)  furnishes  a  partial  check  on  the  computations. 
An  application  of  these    formulae   may  be   found  in 
§  75,  (a). 


CHAPTER  II 
TIME 

11.  The  passage  of  any  point  of  the  celestial  sphere 
across  the  meridian  of  an  observer  is  called  the  transit,  or 
culmination,  or  meridian  passage  of  that  point.  In  one 
rotation  of  the  sphere  about  its  axis,  every  point  of  the 
sphere  is  twice  on  the  meridian;  once  at  upper  culmina- 
tion (above  the  pole),  and  once  at  lower  culmination 
(below  the  pole).  For  an  observer  in  the  northern  hemi- 
sphere, a  star  whose  north  polar  distance  is  less  than  the 
latitude  is  constantly  above  the  horizon,  and  both  culmi- 
nations are  visible;  a  star  whose  south  polar  distance  is 
less  than  the  latitude  is  constantly  below  the  horizon,  and 
both  culminations  are  invisible;  and  a  star  between  these 
limits  is  visible  at  upper  culmination,  but  invisible  at  the 
lower.  For  an  observer  in  the  southern  hemisphere  the 
first  two  cases  are  reversed. 

Three  systems  of  time  are  required  in  the  operations  of 
practical  astronomy:  sidereal,  apparent  (or  true)  solar  and 
mean  solar. 

A  sidereal  day  is  the  interval  of  time  between  two  suc- 
cessive transits  of  the  true  vernal  equinox  over  the  same 
meridian.  The  sidereal  time  at  any  instant  is  the  hour 
angle  of  the  vernal  equinox  at  that  instant.  It  is  Oft  Ow  0* 
when  the  vernal  equinox  is  on  the  meridian  —  this  instant 
is  called  sidereal  noon  —  and  is  reckoned  through  24  hours. 
The  sidereal  time  is  also  equal  to  the  right  ascension  of 
the  observer's  meridian,  since  the  right  ascension  of  the 
meridian  is  equal  to  the  hour  angle  of  the  vernal  equinox. 

15 


16  PRACTICAL   ASTRONOMY 

It  follows,  then,  that  any  star  will  be  at  upper  culmina- 
tion at  the  instant  when  the  sidereal  time  is  equal  to  the 
star's  right  ascension;  and  at  lower  culmination  when  the 
sidereal  time  differs  12  hours  from  the  star's  right  ascen- 
sion. The  rotation  of  the  earth  on  its  axis  is  perfectly 
uniform;  but  owing  to  precession  and  nutation  the  vernal 
equinox  has  a  minute  and  irregular  motion  to  the  west 
(amounting  on  the  average  to  0'M26  per  day):  so  that  a 
sidereal  day  does  not  correspond  exactly  to  one  rotation  of 
the  earth,  nor  is  its  length  absolutely  uniform,  but  it  is 
sensibly  so. 

An  apparent  (or  true)  solar  day  is  the  interval  of  time 
between  two  successive  upper  transits  of  the  sun  over  the 
same  meridian.  The  hour  angle  of  the  sun  at  any  instant 
is  the  apparent  time  at  that  instant.  It  is  reckoned  from 
0*  Om  0*  at  noon  —  called  apparent  noon  —  through  24 
hours.  But  the  apparent  day  varies  greatly  in  length, 
for  two  reasons,  viz. : 

First,  —  The  earth  moves  in  an  ellipse  with  a  variable 
velocity.  Hence  the  sun's  (apparent)  eastward  motion 
(in  longitude)  is  variable. 

Second, — The  sun's  (apparent)  motion  is  in  the  ecliptic. 

Hence  the  sun's  motion  in  right  ascension  and  hour 
angle  is  variable,  and  a  clock  cannot  be  rated  to  keep 
apparent  time. 

A  convenient  solar  time  is  obtained  in  this  way : 
Assume  an  imaginary  body  to  move  in  the  ecliptic  with 
a  uniform  angular  velocity  such  that  it  and  the  sun  pass 
through  perigee  at  the  same  instant.  Assume  a  second 
imaginary  body  to  move  in  the  equator  with  a  uniform 
angular  velocity  such  that  the  two  will  pass  through  the 
vernal  equinox  at  the  same  instant.  The  second  body  is 
called  the  mean  sun. 

A  mean  solar  day  is  the  interval  of  time  between  two 
successive  upper  transits  of  the  mean  sun  over  the  same 
meridian.  The  hour  angle  of  the  mean  sun  is  the  mean 


TIME  17 

time.  It  is  reckoned  from  0AOmO*  at  noon  —  called  mean 
noon  —  through  24  hours. 

The  difference  between  the  apparent  and  mean  time  is 
called  the  equation  of  time.  Its  value  is  given  in  the 
American  Ephemeris  for  the  instants  of  Greenwich  appar- 
ent and  mean  noon  and  Washington  apparent  noon, 
whence  its  value  may  be  obtained  for  any  other  instant 
by  interpolation. 

The  astronomical  solar  day  begins  at  noon,  whereas  the 
day  popularly  used  —  called  the  civil  day — begins  at  (the 
preceding)  midnight.  Thus,  Feb.  1,  10A  A.M.,  civil  reck- 
oning, is  Jan.  31rf  22A  astronomical  mean  time. 

12.  The  interval  of  time  between  two  successive  pas- 
sages of  the  mean  sun  through  the  mean  vernal  equinox 
—  called  a  tropical  year  —  was  for  the  year  1800,  accord- 
ing to  Bessel,  365.24222  *  mean  solar  days. 

The  number  of  sidereal  days  in  this  interval  is  366.24222, 
since  in  that  interval  of  time  the  mean  sun  moves  eastward 
through  about  360°,  and  therefore  the  vernal  equinox  dur-< 
ing  the  year  makes  one  more  transit  over  any  given  merid- 
ian than  the  sun.     Thus  we  have 

365.24222  mean  days  =  366.24222  sidereal  days. 
Whence 

24*  mean  time    =  24A  3m  568.555  sidereal  time, 
24*  sidereal  time  =  23  56       4.091  mean  time. 

From  these  equations  it  is  found  that  the  gain  of  side- 
real time  on  mean  time  in  one  mean  hour  is  9*.  8565;  and 
in  one  sidereal  hour,  95.8296.  These  are  the  amounts  by 
which  the  right  ascension  of  the  mean  sun  increases  in  one 
mean  and  one  sidereal  hour,  respectively. 

*  The  length  of  the  tropical  year  is  diminishing  at  the  rate  of  about  0».6 
per  century.     This  is  due  to  the  fact  that  the  mean  vernal  equinox  is 
moving  westward  at  an  accelerated  rate,  as  will  be  seen  later,  from  the 
last  of  equations  (120). 
c 


18  PRACTICAL    ASTRONOMY 

CONVERSION    OF   TIME 

13.  In  nearly  every  problem  of  practical  astronomy  it 
is  necessary  to  convert  the  time  at  one  place  into  the  cor- 
responding time  at  another  place,  or  to  convert  the  time 
in  one  system  into  the  corresponding  time  in  another  sys- 
tem.    By  means  of  the  data  furnished  in  the  Ephemeris 
this  is  readily  done. 

14.  To  convert  the  time  at  one  place  into  the  correspond- 
ing time  at  another. 

Since  every  epoch  of  time  is  defined  by  an  hour  angle, 
the  difference  of  time  at  two  places  is  the  difference  of 
the  two  corresponding  hour  angles ;  and  that  is  equal  to 
the  difference  of  the  longitudes  of  the  two  places.  There- 
fore, if  the  difference  of  longitude  be  added  to  the  time  at 
the  western  place  the  sum  is  the  corresponding  time  at 
the  eastern.  If-it  be  subtracted  from  the  time  at  the  east- 
ern place  the  result  is  the  time  at  the  western. 

Example  1.  The  Ann  Arbor  mean  time  is  1891  March 
I0d  21A  10m  54*.70.  What  is  the  corresponding  Greenwich 
mean  time  ? 

Ann  Arbor  mean  time,  1891  March  10d  21*  10™  54«.70 

Longitude  Ann  Arbor,  Amer.  Ephem.,  p.  482,      +  5  34    55  .14 
Greenwich  mean  time  1891  March  11     2  45    49.84 

Example  2.  The  Washington  sidereal  time  is  Oft  23m 
17MO.  What  is  the  corresponding  Ann  Arbor  sidereal 
time? 

Washington  sidereal  time,  0*  23*  17'.10 

Difference  of  longitude,  Amer.  Ephem.,  p.  482,         0  26    43  .10 
Ann  Arbor  sidereal  time,  23   56    34  .00 

Example  3.  The  Ann  Arbor  apparent  time  is  1891 
March  20<*  21A  58m  19M7.  What  is  the  Berlin  apparent 
time  at  the  same  instant  ? 

Ann  Arbor  apparent  time,  1891  March  20d  21*  58™  19*.  17 

Difference  of  longitude,  6  28    30  .05 

Berlin  apparent  time,  1891  March  21     4  26    49  .22 


CONVERSION    OF   TIME  19 

15.  To  convert  apparent  time  at  any  place  into  mean 
time,  and  vice  versa. 

The  equation  of  time  at  the  given  instant  is  required. 
When  this  is  applied  with  the  proper  sign  to  the  one,  it 
gives  the  other.  If  apparent  time  is  given,  convert  it  into 
Greenwich  apparent  time,  and  take  the  equation  of  time 
from  page  I  of  the  given  month  in  the  Ephemeris.  If 
mean  time  is  given,  convert  it  into  Greenwich  mean  time, 
and  take  the  equation  of  time  from  page  II  of  the  month. 

In  taking  these  and  other  data  from  the  Ephemeris, 
care  must  be  exercised  in  making  the  interpolations. 
Thus,  let  it  be  required  to  determine  the  equation  of  time 
at  Greenwich  apparent  time  1891  Feb.  24d  10*.  Its  value 
for  apparent  noon  is  -f-  13W  25s.  52,  and  the  difference  for 
one  hour  at  noon  of  that  day  is  05.381.  The  difference 
for  one  hour  at  noon  the  next  day  is  0s. 406.  The  hourly 
difference  is  therefore  variable,  but  we  may  assume  the 
second  difference  to  be  constant.  The  change  in  the  equa- 
tion during  the  10  hours  after  noon  is  ten  times  the  average 
hourly  change  for  the  10  hours  ;  that  is,  since  the  second 
difference  is  constant,  ten  times  the  hourly  change  at  the 
middle  period,  or  at  5  hours  after  noon.  The  average 
hourly  change  is  0s.  386,  and  the  desired  equation  of  time  is 
+  13™  25*.52  -  10  x  (K386  =  +  13"  21-.66. 

Example.  The  Berlin  mean  time  is  1891  Feb.  28<*  0* 
II*1  20s.  60.  What  is  the  apparent  time  ? 

Berlin  mean  time,  1891  Feb.  28d  0*  llw  20«.60 

Longitude  Berlin,  0  53    34  .91 

Greenwich  mean  time,  Feb.  27  23  18 

This  is  23*.  30  after  Gr.  mean  noon  Feb.  27,  or  0A.70 
before  noon  Feb.  28.     In  this  and  similar  cases  the  inter- 
polation should  be  made  for  the  interval  before  noon. 
Equation  of  time,  Gr.  mean  noon,  Feb.  28,  -  12"»  44'.52 

Change  before  noon,  0.70  x  0«.473,  0  .33 

Equation  of  time,  —  12    44  .85 

Berlin  apparent  time,  1891  Feb.  27d  23*  58    35 .75 


20  PRACTICAL   ASTRONOMY 

16.  To  convert  a  mean  time  interval  into  the  equivalent 
sidereal  interval,  and  vice  verm. 

In  §  12  it  is  shown  that  sidereal  time  gains  9s.  8565 
on  mean  time  in  one  mean  hour.  The  corresponding 
gain  for  any  number  of  hours,  minutes,  and  seconds,  is 
tabulated  in  Table  III  of  the  appendix  to  the  American 
Ephemeris.  If  this  gain  be  added  to  the  mean  time  inter- 
val, the  sum  is  the  equivalent  sidereal  interval. 

The  gain  of  sidereal  time  on  mean  time  in  one  sidereal 
hour  is  9s.  8296.  The  corresponding  gain  for  any  number 
of  hours,  minutes,  and  seconds,  is  tabulated  in  Table  II 
of  the  appendix  to  the  American  Ephemeris.  If  this 
gain  be  subtracted  from  the  sidereal  interval,  the  differ- 
ence is  the  equivalent  mean  time  interval. 

Example  1.  A  mean  time  interval  is  llh  33m  215.76. 
Find  the  corresponding  sidereal  interval. 

Mean  time  interval,  17*  33™  2K76 

Gain  of  sidereal  on  mean,  Table  III,  2    53  .04 

Sidereal  interval,  17   36    14  .80 

Example  2.  A  sidereal  time  interval  is  17*  36W  14s.  80. 
Find  the  corresponding  mean  time  interval. 

Sidereal  interval,  17*  36™  14*. 80 

Gain  of  sidereal  on  mean,  Table  II,  2    53  .04 

Mean  time  interval,  17  33    21 .76 

17.  To  convert  mean  time  into  sidereal  time. 

Mean  time  at  any  instant  is  the  interval  after  mean 
noon.  If  this  interval  be  converted  into  the  equivalent 
sidereal  interval  and  added  to  the  sidereal  time  at  noon, 
the  sum  will  be  the  sidereal  time  required.  The  sidereal 
time  at  noon  is  equal  to  the  right  ascension  of  the  mean 
sun  at  this  instant.  The  Ephemeris  gives  on  page  II  for 
the  month  the  sidereal  time,  or  the  right  ascension  of  the 
mean  sun,  at  Greenwich  mean  noon,  whence  its  right 


CONVERSION   OF   TIME  21 

ascension  at  noon  for  a  place  whose  longitude  is  L  may 
be  obtained  by  applying  the  term  L  x  9*.  8565,*  from 
Table  III  of  the  appendix  to  the  American  Ephemeris. 

Example.  The  Ann  Arbor  mean  time  is  1891  Feb.  20d 
11*  45™  20*.40.  What  is  the  equivalent  sidereal  time  ? 

Right  ascension  mean  sun  at  Gr.  mean  noon,  Feb.  20,  22*  Om  31*.75 

.  'Change  in  5*  34"  55s.  14,  Table  III,  0  55  .02 

Right  ascension,  or  sid.  time,  at  Ann  Arbor  mean  noon,  22  1  26  .77 

Mean  time  interval  after  noon,  11  45  20  .40 

v  Gain  of  sidereal  on  mean  time,  Table  III,  1  55  .87 

Equivalent  sidereal  interval  after  noon,  11  47  16  .27 

Sidereal  time,  9  48  43.04 

For  Ann  Arbor,  and  similarly  for  other  stations,  the 
quantity  L  x  9*.  8565  =  55s.  02  is  a  constant,  and  is  held 
in  mind  by  the  computer. 

Likewise,  the  experienced  computer  writes  down  the 
four  necessary  quantities  and  combines  them  all  in  one 

addition,  thus : 

22*   0™31*.75 
55.02 

11  45    20.40 
1    55.87 

9  48    43.04 

18.    To  convert  sidereal  time  into  mean  time. 

If  the  sidereal  time  at  the  preceding  mean  noon  (formed 
as  before)  be  subtracted  from  the  given  time,  the  result 
is  the  sidereal  interval  after  mean  noon.  This  interval 
converted  into  the  equivalent  mean  time  interval  is  the 
mean  time  desired. 

Example.  On  1891  Feb.  20,  the  sidereal  time  at  Ann 
Arbor  is  9*  48m  43'.  04.  What  is  the  mean  time  ? 

*  It  must  be  remembered  that  for  a  station  east  of  Greenwich  the 
quantity  L  x  9«.8565  is  negative. 


22  PRACTICAL   ASTRONOMY 

Right  ascension  mean  sun  at  Gr.  mean  noon,  Feb.  20,   22*    Ow  3K75 

Change  in  5*  34m  55«.14,  Table  III,  0  55  .02 

Right  ascension,  or  sid.  time,  at  Ann  Arbor  mean  noon,  22  1  26  .77* 

The  given  sidereal  time,                                                        9  48  43  .04 

Sidereal  interval  after  mean  moon,  Feb.  20,                     11  47  16  .27 

Gain  of  sidereal  on  mean  time,  Table  II,  1  55  .87 

Ann  Arbor  mean  time,                                        Feb.  20d  11  45  20  .40 

It  should  be  noted  that  in  the  original  statement  of  this 
example,  the  date  1891  Feb.  20  is  the  astronomical  mean 
solar  date.  The  observer  should  always  record  this  date 
with  care,  especially  in  the  case  of  observations  taken  near 
noon,  as  ambiguity  may  otherwise  arise.  It  would  be 
well  in  such  cases,  as  indeed  in  all  cases,  to  record  also 
the  civil  day  of  the  week.f  Thus  the  statement  UA  day- 
light meteor  was  observed  at  Ann  Arbor,  1891  Dec.  21, 
at  sidereal  time  187*  2m  30s,"  is  ambiguous  to  the  extent  of 
one  sidereal  day,  since  on  that  solar  day  the  sidereal  time 
was  twice  equal  to  ~L8h  2m  305.  The  record  may  refer  to  a 
phenomenon  observed  just  after  noon  of  Monday  or  just 
before  noon  of  Tuesday.  If  the  record  were  written 
"  Monday,  1891  Dec.  20  "  there  would  be  no  uncertainty. 

*  This  quantity  is  to  be  subtracted  from  the  one  directly  following. 

t  Inasmuch  as  one  may  easily  record  an  erroneous  day  of  the  month, 
many  observers  have  the  admirable  practice  of  beginning  their  records 
with  the  day  of  the  week.  Thus,  "  Wednesday,  1891  July  8  "  suffices  for 
observations  made  on  Wednesday  afternoon,  or  for  continuous  observa- 
tions throughout  Wednesday  night ;  but  an  isolated  observation  made 
the  next  morning  may  be  headed,  "Thursday  morning,  1891  July  8." 


\ 


CHAPTER   III 


CORRECTION  OF  OBSERVATIONS 

19.  The  observed  directions  of  all  bodies  in  the  solar 
system  are  sensibly  different  for  observers  at  different 
places  on  the  earth's  surface.     These  differences  must  be 
allowed  for  before  observations  made  at  different  places 
can  be  compared.     This  is  accomplished  by  reducing  all 
observations  to  the  center  of  the  earth,  to  which  point  the 
data  of  the  Ephemeris  refer.     A  knowledge  of  the  form 
and  size  of  the  earth  is  therefore  indispensable. 

FORM    AND    DIMENSIONS    OF   THE   EAKTH 

20.  Geodetic  measurements,  combined  with  astronomical 
observations,  have  shown  that  the  earth  is  very  nearly  an  ob- 
late spheroid  whose  minor  axis  coincides  with  the  polar  axis. 

Let   QPQ'P1  be  an  elliptical  section  of  the  spheroid 
made  by  the  meridian  of  an  observer  at  0  ;  A  the  center 
of  the  earth  ;  N8  the  horizon  ;  and  let 
a    =  semi-major   axis   of   ellipse 


b    =  semi-minor   axis   of  ellipse 

=  AP, 
<f>  =  geographical  latitude*  of  0 


<f>'  =  geocentric     latitude     of     0 


p    =  radius  of  earth  at  O  =  AO, 
<£'  —  <£  —  reduction  to   geocentric 

latitude  =  A  OB, 
x,  y  =  rectangular  coordinates  of 

0  =±  A  C,  CO. 


Q 


FIG.  2 


*  It  frequently  happens,  especially  in  mountainous  regions,  that  the 
plumb-line  is  not  normal  to  the  theoretical  ellipsoidal  surface  of  the  earth, 

23 


24  PRACTICAL   ASTRONOMY 

From  a  discussion  of  all  available  observations,  Bessel 

found 

a  =  3962.802  miles,  b  =  3949.555  miles; 

and  therefore,  for  the  eccentricity  of  a  meridional  section, 
QPQ'P', 

e  =  0.0816967. 

21.  Given  the  geographical  latitude  of  a  point  on  the 
earths  surface,  required  the  corresponding  geocentric  lati- 
tude. 

The  equation  of  the  ellipse  (Fig.  2)  is 


Differentiating,  and  substituting 

tan  <£  =  -—,  tan  <£>'  =  #, 

dy  x 

we  obtain  the  desired  relation 

tan  <£'  =  —^  tan  <£  =  (!-  e'2)  tan  <£•  (5?) 

The  reduction  to  the  geocentric  latitude,  <f>r  —  <£,  can 
be  expressed  in  terms  of  <f>.     If  the  equation 

tan  x  =  p  tan  y, 

which  is  identical  in  form  with  (57),  be  developed  in 
series  it  becomes  [Chauvenet's  Plane  Trig.,  §  254] 

x  -  y  =  q  sin  2  y  +  £  ^2  sin  4  y  +  $qs  sin  6  y  +  •••  , 

in  which 


owing  to  the  fact  that  the  local  irregularities  of  surface  and  of  density  be- 
come appreciable.  In  such  cases,  the  zenith  determined  by  the  plumb- 
line  will  not  coincide  with  the  theoretical  zenith.  Consequently  the 
latitude  and  longitude,  determined  astronomically,  will  differ  from  the 
latitude  and  longitude  determined  geodetically.  The  geodesist  has  to 
deal  with  both  systems,  but  the  astronomer  uses  only  the  former. 


PARALLAX  25 

Substituting  from  (57)  the  values  corresponding  to  x, 
y,  and  jt?,  and  dividing  by  sin  1"  in  order  to  express  the 
result  in  seconds  of  arc,  we  obtain  the  practically  rigorous 

formula 

<£'  _  <£  =  _  690".65  sin  2  <j>  +  I'M  6  sin  4  <£.  (58) 

22.    To  find  the  radius  of  the  earth  for  a  given  latitude. 
Substituting  x  =  p  cos  <f>f  and  y  —  p  sin  <£'  in  (56)  and 
eliminating  b  by  (57),  we  obtain 


" — -  ')cos  ,•  <59> 

In  using  this  equation  make  a  =  1,  since  the  equatorial 
radius  is  taken  as  the  unit. 

The  values  of  <£'  —  <f>  and  of  p  for  the  positions  of  the 
principal  observatories  are  given  on  pp.  482-485  of  the 
American  Ephemeris  for  1891. 

Formulae  (58)  and  (59)  give  the  correct  values  of 
<f>'  —  (f>  and  p  at  sea  level.  It  is  evident  that  the  alti- 
tude of  the  observer  above  sea  level  must  be  taken  into 
account.  The  slight  corrections  thus  rendered  necessary 
may  be  computed  from  elementary  principles  of  trigo- 
nometry. 

PARALLAX 

23.  The  geocentric  or  true  place  of  a  star  is  that  in 
which  it  would  be  seen  by  an  observer  at  the  center  of  the 
earth.  The  apparent  *  or  observed  place  is  that  in  which 
it  is  seen  by  the  observer  on  the  surface  of  the  earth. 
The  parallax  of  a  star  is  the  difference  between  its  true 
and  apparent  places.  It  may  also  be  defined  as  the  angle 
at  the  star  subtended  by  the  radius  of  the  earth  drawn  to 
the  point  of  observation.  This  angle  is  approximately  a 

*  The  terms  true  and  apparent  are  used  in  a  relative  sense  only.  In 
reference  to  parallax,  the  true  place  is  the  place  corrected  for  parallax. 
In  reference  to  refraction,  the  apparent  place  is  affected  by  refraction,  the 
true  place  is  corrected  for  refraction  ;  and  similarly  in  other  subjects. 


26  PRACTICAL   ASTRONOMY 

maximum  for  an  observer  at  a  given  place  when  the  star  is 
seen  in  his  horizon.  It  is  then  called  the  horizontal  paral- 
lax. When  the  observer  is  at  a  place  on  the  earth's  equator 
this  angle  is  called  the  equatorial  horizontal  parallax. 

24.  To  find  the  equatorial  horizontal  parallax  of  a  body. 

In  Fig.  3  let  8'  be  a  body  in 
the  horizon  of  a  point  0  on  the 
earth's  equator.  Then  if 

a  =  equatorial  radius  of  the  earth 

=  CO, 
A  =  body's  distance  from  the  earth's 

center  =  CSf, 

TT  =  equatorial    horizontal   parallax 
FIG.  3  =CS'0, 

we  have  sin  TT  =  —  •  (60) 

The  astronomical  unit  of  distance  is  the  mean  distance 
of  the  earth  from  the  sun.  Using  (60)  with  a  and  A 
expressed  in  terms  of  this  unit,  the  American  Ephemeris 
tabulates  the  values  of  TT  for  the  moon  [page  IV  of  the 
month]  and  the  planets  [pp.  218-249] ;  employing  values 
of  a  and  the  earth's  mean  distance  from  the  sun  such  that 
the  sun's  mean  equatorial  horizontal  parallax  is  8". 848. 

Recent  researches  have  shown  that  8 ".80  is  probably  a 
much  more  correct  value  of  the  sun's  mean  equatorial 
horizontal  parallax,  and  the  superintendents  of  the  prin- 
cipal astronomical  annuals  have  agreed  to  use  that  value  in 
computing  ephemerides  from  about  the  year  1900. 

25.  To  find  the  parallax  in  zenith  distance,  the   earth 
being  regarded  as  a  sphere. 

In  Fig.  3  let 

z'  —  apparent  zenith  distance  of  a  star  S  =  ZOSt 

z  =  true  zenith  distance  =  ZCS, 

p  =  parallax  in  zenith  distance  =  CSO  =  z'  —  z. 


PARALLAX 

Then  from  the  triangle  COS 


27 


sin  p      a 

=£-5  =  -  =  sin  TT, 

sm  z      A 


or 


sin  p  —  sin  TT  sin  z'. 
For  all  bodies  except  the  moon  we  can  write 

p  =  TT  sin  z'. 
For  the  sun  we  have,  from  (60), 


(61) 
(62) 


and  (62)  becomes 


(63) 


The   values   of  log  A   are   tabulated   in   the   Ephemeris, 
page  III  of  the  month. 

In  the  case  of  observations  made  with  a  sextant,  sur- 
veyor's transit,  or  other  similar  instrument,  we  may 
assume  A  equal  to  unity,  and  take  the  required  value  of  p 
from  the  following  table  computed  from 

:  (64) 


z' 

P 

z1 

P 

0° 

0".0 

50° 

6".7 

10 

1  .5 

60 

7  .6 

20 

3  .0 

70 

8  .3 

30 

4  .4 

80 

8  .7 

40 

5  .7 

90 

8  .8 

For  refined  observations  (61)  is  not  sufficiently  exact, 
and  recourse  must  be  had  to  formulae  which  consider  the 
earth  as  a  spheroid. 

26.  Griven  the  true  zenith  distance  and  azimuth  of  a  star, 
required  its  apparent  zenith  distance  and  azimuth,  the  earth 
being  regarded  as  a  spheroid. 


28  PRACTICAL   ASTRONOMY 

Let  the  star  be  referred  to  a  system  of  rectangular  axes 
whose  origin  is  at  the  point  of  observation,  the  positive 
axis  of  x  being  directed  to  the  south  point,  the  positive 
axis  of  y  to  the  west  point  and  the  positive  axis  of  z  to 
the  zenith.  Let 

X',  Y'j  Z'  =  the  rectangular  coordinates  of  the  star, 
A'  =  the  star's  distance  from  the  observer, 
A'  =  its  apparent  azimuth, 
z'  —  its  apparent  zenith  distance. 
Then 

X'  =  A'  sin  z'  cos  A', 

P  =  A'  sin  z'  sin  A', 
Z'  =  A'  cos  z'. 

Again,  let  the  star  be  referred  to  a  second  system  of 
rectangular  axes  parallel  to  the  first,  the  origin  being  at 
the  center  of  the  earth.  Let 

X,  Y,  Z  =  the  rectangular  coordinates  of  the  star, 

A  =  the  star's  distance  from  the  origin,  0 

A  =  its  true  azimuth, 

z  =  its  true  zenith  distance. 
Then 

X  =  A  sin  z  cos  A, 

Y  =  A  sin  z  sin  A, 
Z  —  A  cos  z. 

Let  the  coordinates  of  the  point  of  observation  referred 
to  the  second  system  be  X",  Y",  Z'1.  From  Fig.  2  it  is 
seen  that 

X"  =  p  sin  (<£  -  <£'),     Y"  =  0,     Z"  =  p  cos  (<£  -  <£')• 
Now 

X'  =  X  -  X",     T  =  Y  -  Y",    Z'  =  Z  -  Z", 

and  therefore 

A'  sin  z'  cos  A'  —  A  sin  z  cos  A  —  p  sin  (<£  —  <£'),  ^ 

A'  sin  z'  sin  A'  =  A  sin  z  sin  A,  (65) 

A'  cos  z'  =  A  cos  z  —  p  cos  (<£  —  <£').  > 


PARALLAX  29 

These  equations  completely  determine  A',  zf  and  Ar,  and 
therefore  the  parallax  z'  —  z  and  A'  —A.  It  'is  better, 
however,  to  transform  them  so  that  the  parallax  can  be 
computed  directly.  For  this  purpose,  divide  the  equations 
through  by  A  and  put 


also  substitute  from  (60),  a  being  unity, 


and  we  have 

/sin  z'  cos  A'  =  sin  z  cos  A  —  p  sin  TT  sin  (<j>  —  <£'),  (66) 

/  sin  z'  sin  A  '  —  sin  z  sin  A  ,  (67) 

/cos  2'  =cosz  —  psiiiTrcos  (<£  —  <£').  (68) 

From  (66)  and  (67)  we  obtain 

/  sin  z'  sin  (A'  -  A  )  =  p  sin  TT  sin  (<£  -  <£')  sin  4  ,  (69) 

/sin  z'  cos  (^4'  —  ^4)  =  sin  z  —  p  sin  TT  sin  (<j>  —  <£')  cos  A.      (70) 

Putting 

m  =  p  sin  TT  sin  (<£-<£') 

sin  2 
(69)  and  (70)  give 


tan  (A'  -  A)  =  -  (72) 

1  -  m  cos  A 

.  Multiplying  (69)  by  sin  J  (A!  -  ^.)  and  (70)  by 
cos  J  (J.'  —  A),  adding  the  products  and  dividing  by 
cos  J  (A!  —  J.),  we  obtain 

/sin  z'  =  smz-p  sin  TT  sin(<#>  -  <£')  COSK^^  +  ^).          (73) 

cos 

Let  us  assume 

then 

/sin  2'  =  sin  z  —  p  sin  TT  cos  (<^  —  <£')  tan  y.  (75) 

This  combined  with  (68)  gives 


30  PRACTICAL   ASTRONOMY 

/sin  (z'  -  z)  =         p  sin  TT  cos  (  <ft  -  <£')  sm  (2  ~  ?),         (76) 

cosy 

/cos  (2'  -  z)  =  1  -  p  sin  TT  cos  (<£  -  <£')  cos  C*  -  y).         (77) 

cosy 

Assume 

n  _  p  sin  TT  cos  (<fr  -  <fr') 
cosy 

and  we  have 


tan  (z'  -z)  =  -          .  (79) 

1  -  n  cos  (z  -  y) 

Formulae  (71)  and  (72)  rigorously  determine  the  par- 
allax in  azimuth,  and  (74),  (78)  and  (79)  the  parallax  in 
zenith  distance.  We  may  abbreviate  the  computation  by 
writing  (74)  in  the  form 

y  =  (<£-<£')  COS  .4,  (80) 

which  is  in  all  cases  sufficiently  exact. 

27.  (riven  the  apparent  zenith  distance  and  azimuth  of  a 
body,  required  its  true  zenith  distance  and  azimuth,  the  earth 
being  regarded  as  a  spheroid. 

From  (68)  and  (75)  we  obtain 

sin  (z'  -  z)  =  P  si"  *  cos  (<fr  -  <£')  sin  (*'  -  Y)^ 
cosy 

for  which,  since  <£  —  <£'  and  7  are  small  angles,  we  can  / 

write 

sin  (z'  —  z)  =  p  sin  TT  sin  (z'  —  y),  (81) 

in  which  7  is  given  without  sensible  error  by 

y  =  (^_^)cos^'.  (82) 

We  obtain  from  (66)  and  (67) 

sin  (A*  -A)=  p  sin  TT  sin  Oft  -<£')  sin  ^  (g3) 

sin  2 

in  which  the  value  of  z  is  found  by  the  solution  of  (81). 
Formulae  (82),  (81)  and  (83)  completely  solve  the  prob- 
lem. For  all  known  bodies  save  the  moon  we  may  write 


PARALLAX  31 

z'  -  z  =  p  TT  sin  ( sf  -  y),  (84) 

A'  -  A  =  p  TT  sin  (<£  -  <£')  sin  4'  cosec  zf.  (85) 

An  application  of  the  formulae  of  this  section  will  be 
;ound  in  §  89,  in  the" determination  of  longitude  by  lunar 
distances. 

28.  To  find  the  parallax  of  a  body  in  right  ascension  and 
declination.  Let 

a  =  the  body's  geocentric  right  ascension, 


8  = 

t  = 

A  = 

/ 

CL  — 

8'  = 

t'  - 


"          declination, 
"          hour  angle, 
"          distance, 
apparent  right  ascension, 
"          declination, 


"  "          hour  angle, 

"  "          distance, 

0  =  the  observer's  sidereal  time. 

By  methods  similar  to  those  used  in  developing  equa- 
tions (72),  (74)  and  (79),  we  may  obtain  the  correspond- 
ing equations, 

p  sin  TT  cos  <f>'  sin  t 

tan  (a  -  a')  = £ — t  (86) 

cos  o  —  p  sin  TT  cos  <f>'  cos  t 

tan  <f>'  cos  i  (a  —  a') 

tan  y  = E 1_^_ L,  (87) 

7      cos  [0  -  \  (a  +  a')] 


tan  (8  -  8')  =  - 

sin  y  —  p  sin  TT  sin  <p  cos  (y  —  6) 

These  rigorously  determine  the  parallax  in  right  ascen- 
sion, a  —  a',  and  the  parallax  in  declination,  8  —  S;,  when 
the  geocentric  coordinates  are  the  known  quantities.  If 
the  apparent  coordinates  a',  Sf  and  t'  have  been  obtained 
by  observation,  and  a,  S  and  t  are  unknown,  we  substitute 
a',  £'  and  £'  for  a,  8  and  £  in  the  second  members,  and 
solve.  The  resulting  approximate  values  of  the  parallax 
furnish  nearly  correct  values  of  a,  S  and  t.  Employing 
these  in  a  second  solution  of  the  equations  we  shall  obtain 
sufficiently  exact  values  of  the  parallax. 


32  PRACTICAL    ASTRONOMY 

29.  For  all  known  bodies  except  the  moon  the  values 
of  TT,  a  —  a1  and  S  —  8'  will  be  very  small,  and  we  may 
write  (86),  (87)  and  (88),  without  sensible  error,  in  the 
form 


A  •  cos  8 


(88) 


'tan  y  =          *  ,  (90) 

cos  t 

>_y=8".8P8in»rin(y-8)[ 

A  •  sin  y 

ill  which  A  is  expressed  in  terms  of  the  astronomical  unit* 
of  distance.  These  formulae  will  determine  the  parallax 
satisfactorily  also  if  t  and  8  are  replaced  by  tf  and  £',  for 
which  case  an  application  of  them  will  be  found  in  §  155. 
At  the  fixed  observatories  it  is  customary  to  construct 
tables  which  greatly  facilitate  the  computation  of  paral- 
laxes. The  equations  (89)  and  (91)  may  be  written 

(a  -  a')  A  =  8".8  p  cos  <j>'  sin  t'  sec  8',  (92) 

(8  -  8')  A  =  8".8  p  sin  <£'  cosec  y  sin  (y  -  8')-  (93) 

The  second  members  of  these  equations  are  the  parallaxes 
in  right  ascension  and  declination  of  an  imaginary  body  at 
distance  unity  when  observed,  at  a  given  station  (/o,  (//), 
in  the  direction  t',  8'.  They  are  called  parallax  factors. 
Their  values  are  generally  computed  and  tabulated,  at  a 
given  observatory,  for  every  10™  of  hour  angle  and  every 
degree  of  declination.  When  a  body  is  observed  at  any 
hour  angle  and  declination,  the  corresponding  parallax 
factors  may  be  obtained  by  interpolation  from  the  tables. 
The  parallaxes,  a  —  a'  and  S  —  8',  may  then  be  determined 
by  dividing  the  parallax  factors  by  the  distance  A  of  the 
body,  as  will  be  seen  from  (92)  and  (93).  An  application 
of  these  formulse  will  be  found  in  §  154. 

REFRACTION 

30.    It  is  shown  in   Optics  that  when  a  ray  of  light 
passes  obliquely  from  one  transparent  medium  into  an- 


REFRACTION 


33 


other  of  greater  density,  it  is  refracted  from  its  original 
direction  according  to  the  following  laws  : 

(a)  The  incident  ray,  the  normal  to  the  surface  which 
separates  the  two  media  at  the  point  of  incidence,  and  the 
refracted  ray,  lie  in  the  same  plane. 

(6)  .The  sines  of  the  angles  of  incidence  and  refraction 
are  inversely  as  the  indices  of  refraction  of  the  two  media. 

A  ray  of  light  coming  from  a  star  to  an  observer  is 
assumed  to  travel  in  a  straight  line  until  it  reaches  the 
upper  limit  of  the  earth's  atmosphere.  It  then  passes 
continually  from  a  rarer  to  a  denser  medium  until  it 
reaches  the  earth's  surface.  If  we  regard  the  earth  as  a 
sphere,  it  follows  .from  (a)  and  (6)  that  the  path  of  the 
ray  is  a  curve  whose  direction  constantly  approaches  the 
center  of  the  earth. 

Let  Fig.  4  represent  a  section  of  the  earth  and  its  atmos- 
phere made  by  a  vertical  plane 
passing  through  the  point  of 
observation  0  and  a  star  S. 
The  path  of  the  ray,  S  a  b  .  .  n  .  . 
0,  lies  wholly  in  this  plane  and 
is  concave  towards  the  earth. 
The  apparent  direction  of  the 
star  is  OS',  a  tangent  to  the 
curve  at  the  point  of  observa- 
tion. The  true  direction  is  that 
of  a  straight  line  joining  0 
and  S.  The  difference  of  these 
directions  is  the  refraction.  It 
appears  that  refraction  increases 
the  altitude,  and  decreases  the  zenith  distance,  of  a  star, 
but  in  general  does  not  affect  its  azimuth.* 

*  It  is  known  that  appreciable  deviations  in  the  azimuth  are  sometimes 
produced  by  refraction,  especially  in  observations  made  very  near  the  hori- 
zon ;  but  as  they  are  due  to  abnormal  and  unknown  arrangements  of  the 
strata  of  air,  there  is  unfortunately  no  direct  method  of  eliminating  them. 


FIG.  4 


34  PBACTICAL   ASTRONOMY 

The  amount  of  the  refraction  depends  upon  the  density 
of  the  air,  which  is  a  function  of  the  atmospheric  pressure 
and  temperature.  Our  knowledge  of  the  state  of  the 
atmosphere  is  very  imperfect.  The  theory  of  refraction 
is  complex  and  tedious,  refraction  tables  to  be  reliable 
must  be  largely  empirical,  and  we  shall  not  attempt  an 
investigation  of  the  subject. 

The  Pulkowa  Refraction  Tables  given  in  the  Appendix, 
TABLE  I,  are  based  on  the  formula 

r  =  fj.  tan  z  (B  T)A  y*  <rf,  (94) 

in  which  z  is  the  apparent  zenith  distance,  /*,  A,  X,  and  a-  are 
functions  of  the  apparent  zenith  distance,  B  depends  on  the 
reading  of  the  barometer,  T  depends  on  the  temperature  of 
the  column  of  mercury  as  indicated  by  the  attached  *  ther- 
mometer, 7  depends  on  the  temperature  of  the  atmosphere 
as  indicated  by  the  external  thermometer,  i  depends  on  the 
time  of  the  year,  and  r  is  the  refraction  in  seconds  of  arc. 

For  logarithmic  computation  (94)  takes  the  form 
log  r  =  log  n  4-  log  tan  z  +  A  (log  B  +  log  T7)  +  X  log  y  +  i  log <r.    (95) 

Observations  should  not  be  made  at  a  greater  zenith 
distance  than  82°  30',  beyond  which  the  amount  of  the 
refraction  is  uncertain.  We  can  compute  an  approximate 
value  of  the  refraction,  however,  by  means  of  the  Supple- 
ment to  TABLE  I,  which  tabulates  the  values  of  log  JJL  tan  z. 

Example.  Given  the  apparent  zenith  distance  81°  11'  0", 
Barom.  29.420  inches,  Attached  Therm.  +  46°.5  F.,  Ex- 
ternal Therm.  +  22°. 3  F.,  time  May  20,  required  the  true 
zenith  distance. 


log£ 

-  0.00260 

log  j* 

1.74132 

logT 

-  0.00055 

log  tan  z 

0.80937 

logBT 

-  0.00315 

A  logBT 

9.99683 

A 

1.0053 

A  logy 

0.02456 

logy 

-f  0.02337 

i  log  a- 

9.99995 

A. 

1.0510 

logr 

2.57203 

logo- 

0.00026 

r 

6'  13".3 

i 

-0.21 

*  That  is,  attached  to  the  barometer. 


REFRACTION  35 

The  true  zenith  distance  is  therefore  81°  17'  13". 3. 

If  the  true  zenith  distance  is  given  and  the  apparent 
zenith  distance  is  required,  an  approximate  value  of  the 
latter  is  first  found  by  applying  the  mean  refraction,  TABLE 
II,  Appendix,  to  the  true  zenith  distance,  and  then  the 
refraction  is  given  by  (94)  as  before. 

TABLE  II  is  constructed  from  (94)  for  a  mean  state  of 
the  atmosphere,  viz.:  Barom.  29.5  inches,  Att.  Therm. 
50°.0,  and  Ext.  Therm.  50°. 0.  The  factor  a*  is  neglected. 

In  case  no  tables  are  available  an  approximate  value  of 
the  refraction  is  given  by 

983  & 


460  +  t 


tans,  (96) 


in  which  b  is  the  barometer  reading  in  inches,  t  the  tem- 
perature of  the  atmosphere  in  degrees  Fahr.,  and  z  the 
apparent  zenith  distance.*  For  zenith  distances  less  than 
75°  it  represents  the  Pulkowa  refractions  within  a  second 
of  arc,  except  for  extreme  states  of  the  atmosphere.  It  is 
especially  convenient  for  field  work  in  which  an  aneroid 
barometer  is  used. 

When  the  barometer  and  thermometer  have  not  been 
read  a  roughly  approximate  value  of   the  refraction  is 

given  by 

(97) 


in  which  K  is  the  value  of  the  refraction  at  zenith  distance 
45°  for  the  mean  barometer  and  thermometer  readings  at 
the  place  of  observation  ;  but  for  large  zenith  distances 
and  extreme  states  of  the  atmosphere  it  cannot  be  used 
safely.  The  value  of  K  for  sea-level  stations  in  the  tem- 
perate zones  is  about  58". 

*  This  formula  is  due  to  Professor  Comstock  :  The  Sidereal  Messenger, 
April,  1890. 


36  PRACTICAL   ASTRONOMY 

REFR ACTION   IN   RIGHT   ASCENSION   AND   DECLINATION 

31.  The  change  in  zenith  distance  due  to  refraction 
gives  rise  to  corresponding  changes  in  right   ascension 
and  declination.     We  know  the  general  relations  existing 
between  these  coordinates,  whence  the  relations  existing 
between  their  increments  may  be  found  by  differentiation. 
From  (7),  S  and  z  being  the  only  variables,  we  have 

cos  8  dB  =  —  (sin  z  sin  <j>  -f  cos  z  cos  <f>  cos  A  )dz, 
which  reduces  by  means  of  (23)  to 

<ti  =  -  cos  9  dz.  (98) 

Differentiating  (30),  regarding  z,  8  and  t  as  variables,  we 
obtain 

—  sin  z  dz  =  (cos  B  sin  <j>  —  sin  8  cos  <£  cos  <)  d&  —  cos  8  cos  <£  sin  t  dt, 
which  by  (31),  (32)  and  (98)  reduces  to 

cos  8  dt  =  sin  q  dz.  (99) 

But  from  (41)  dt  =  —  da.     Making  this  substitution  and 
replacing  dz  by  the  refraction  r,  (98)  and  (99)  become 

dB  =  -  r  cos  q,  (100) 

da  =  —  r  sin  q  sec  8.  (101) 

These  corrections  reduce  from  the  apparent  to  the  true 
values  of  a  and  B.  If  the  true  place  is  given  and  the 
apparent  place  is  required,  the  signs  of  the  corrections 
must  be  reversed. 

To  compute  r  we  must  know  z.  If  z  and  A  are  given, 
q  is  determined  by  (29) ;  if  t  and  8  are  known,  q  and  z  are 
determined  by  (36)  and  (37). 

DIP   OF   THE   HORIZON 

32.  At  sea  the  altitudes  of  celestial  objects  are  meas- 
ured from' the  visible  sea  horizon.     This  is  below  the  true 


tIP   O^   THE   HORIZON1 


37 


H 


horizon  by  an  amount  depending  on  the  elevation  of  the 
observer's  eye  above  the  surface  of  the  sea. 

Let  Fig.  5  represent  a  section  of  the  earth  made  by  a 
vertical  plane  passing  through  the  eye  of  an  observer  at 
0.  OH'  is  a  line  in  the  visible 
horizon,  OH  is  the  corresponding 
line  in  the  true  horizon,  and  HOH' 
is  the  dip  of  the  horizon.  Let 

x  =  the  height  of  the  eye  above  the 

water  in  feet  =  OA, 
a  =  the  radius  of    the  earth  in  feet 

=  AC, 
D  =  the  dip  of  the  horizon  =  HOH' 

=  OCB. 

We  may  write 

tan0  =  ||  =  v^T^=^7|.  (102) 

X2 

But  —  is  a  very  small  quantity  and  may  be  neglected. 
or 

Tan  D  may  be  replaced  by  D  tan  1".  The  apparent  dip 
is  affected  by  refraction.  The  amount  of  this  refraction 
is  uncertain,  but  an  approximate  value  of  the  true  dip 
is  obtained  by  multiplying  the  apparent  dip  by  the  factor 
0.92.  The  mean  value  of  a  is  20888625  feet.  Introduc- 
ing these  quantities  in  (102)  it  reduces  to 


FIG.  5 


D  =  59" 


(103) 


by  which  amount  the  measured  altitude  must  be  decreased. 

A  convenient  rule,  much  used  by  navigators,  follows 
approximately  from  (103),  thus  : 

The  dip  in  minutes  of  arc  is  expressed  by  the  square 
root  of  the  number  of  feet  that  the  observer's  eye  is  above 
the  water.  To  illustrate,  if  the  observer's  eye  is  30  feet 
above  the  water,  the  dip  is  very  nearly  5'.  5. 

The  dip  must  in  all  cases  be  subtracted  from  the  ob- 
served altitude  in  order  to  obtain  the  true  altitude. 


38 


PRACTICAL   ASTRONOMY 


SEMIDIAMETER 

33.  When  we  observe  a  celestial  body  having  a  well- 
defined  disk,  as  in  the  case  of  the   sun  and  moon,  the 
measurements  are  made  with  reference  to  some  point  on 
the  limb,  and  the  position  of  the  center  is  obtained  by 
correcting  the  observation  for  the  angular  semidiameter 
of  the  body. 

The  geocentric  semidiameters  of  the  sun,  moon  and 
major  planets  are  tabulated  in  the  Ephemeris.  The  appar- 
ent semidiameter  of  the  moon,  however,  is  appreciably 
different  for  different  altitudes,  011  account  of  its  nearness 
to  the  earth,  and  its  value  must  be  determined. 

34.  To  find  the  apparent  semidiameter  of  the  moon. 

Let  Fig.  6  represent  a  section 
H\  made  by  a  plane  passing  through 
the  observer  0,  the  center  of  the 
moon  M,  and  the  center  of  the 
earth  (7,  the  earth  being  considered 
a  sphere.*  Let 

S  =  the  moon's  geocentric  semidiameter 

=  MCB, 
S'  —  the  moon's  apparent  semidiameter 


A  =  the  distance  of  the  moon's  center 
F10-  6  from  the  earth's  center  =  CM, 

A'  =  the  distance  of  the  moon's  center  from  the  observer  =  OM, 
TT    —  the  equatorial  horizontal  parallax  of  the  moon, 
p    =  the  parallax  in  zenith  distance  =  OMC, 
z    =  the  moon's  true  zenith  distance  —  ZCM, 
z'    —  the  moon's  apparent  zenith  distance  =  ZOM. 

Then  we  can  write 

sin  S1  _  ,A  _  sin  (z  +  p)  _  cos  z  sin  p 


sin  S      A' 


_  cos  „  , 


sin  z 


*  The  maximum  error  produced  by  neglecting  the  eccentricity  of  the 
meridian  even  in  the  case  of  the  moon  never  exceeds  0".06. 


SEMIDIAMETEK,  39 

From  (61) 

sin  p  =  sin  IT  sin  z'  ;  (104) 

therefore 


sin  Sr  =  sin  S  (  cos  p  +  sin  TT  cos  2 
V  si 


sn  3 

(104)  and  (105)  furnish  very  nearly  an  exact  solution 
of  the  problem.  For  our  purpose,  and  for  all  ordinary 
observations,  we  can  write 

Sf  =  S  (1  +  sin  TT  cos  z}.  (106) 

35.  To  find  the  contraction  of  any  semidiameter  of  the  sun 
or  moon,  produced  by  refraction. 

The  apparent  disk  of  the  sun  or  moon  is  not  circular, 
since  the  refraction  for  the  lower  limb  is  greater  than 
for  the  center,  and  that  for  the  center  is  greater  than  for 
the  upper  limb.  It  will  be  sufficiently  exact  to  assume 
the  disk  to  be  an  ellipse  whose  center  coincides  with  the 
center  of  the  sun  or  moon. 

The  contraction  of  the  vertical  semidiameter  is  found  by 
taking  the  difference  of  the  refractions  for  the  center  and 
the  upper  or  lower  limb. 

The  contraction  of  the  horizontal  semidiameter  for  all 
zenith  distances  less  than  85°  is  very  nearly  constant  and 
equal  to  about  0".25.  For  our  purpose  it  may  be  neg- 
lected, and  we  shall  not  investigate  the  subject. 

The  contraction  of  any  semidiameter  making  an  angle  q 
with  the  vertical  semidiameter  is  readily  obtained  from  the 
properties  of  the  ellipse.  Thus  let 

a    —  the  horizontal  semidiameter, 
b     =  the  vertical  semidiameter, 
S"  =  the  inclined  semidiameter, 
and  we  have 


S"  sin  q  =  x, 
S"  cos  q  =  y  ; 

whence  Sf'  =  ab  (107) 

Va2  cos2  q  +  b2  sin2  q 


40  PRACTICAL   ASTRONOMY 


ABERRATION 

36.  The  observed  direction  of  a  star  differs  from  its 
true  direction  in  consequence  of  the  motion  of  the  ob- 
server in  space.  The  ratio  of  the  velocity  of  light  to  the 
velocity  of  the  observer  is  finite,  and  a  telescope  changes 

its  position  appreciably  while  a 
ray  of  light  is  passing  from  the 
objective  to  the  eyepiece. 

In  Fig.  7  let  0  be  the  center 
of  the  objective  and  E  the  center 
of  the  eyepiece  of  a  telescope  at 
the  instant  when  a  ray  from  a 
star  S  reaches  the  point  0.  If 
OE'  represent  the  direction  and 
velocity  of  the  ray,  and  AB  rep- 
resent the  direction  of  the  ob- 
server's motion  and  EE'  his 

/  1 , >— B    velocity,  the  telescope  will  be  in 

FlG>  7  the  position  0' E'  when  the  ray 

reaches  E' .  While  the  true  direc- 
tion of  the  star  is  E'  0,  the  apparent  direction  is  E'  0'. 
The  change  in  the  apparent  direction,  OE'  0',  is  called  the 
aberration.  The  star  is  apparently  displaced  toward  that 
point  of  the  celestial  sphere  which  the  observer  is  momen- 
tarily approaching.  To  find  the  amount  of  this  displace- 
ment let  (Fig.  7) 

y    =  BE'  0  =  the  angle  between  the  true  direction  of  the  star  and 

the  line  of  the  observer's  motion, 
y'  =  BE'O'  =  the  angle  between  the  apparent  direction  of  the  star 

and  the  line  of  the  observer's  motion, 
dy  =  y  —  y'  =  the  correction  for  aberration  in  the  plane  of  the  star 

and  the  observer's  motion, 
V  =  OE'      =  the  velocity  of  light, 
v    =  EEf      =  the  velocity  of  the  observer. 


ABERRATION 


41 


Then  from  the  triangle  EOE'  we  have 

sin  (y  —  y7)  =  sin  dy  =  —  sin  •/. 

But  dy  is  always  very  small  and  we  can  write,  without 
sensible  error, 


dy  = 


Fsinl' 


(108) 


which  determines  the  correction  for  aberration  when  t;,  V 
and  7  are  known. 

37.  The  velocity  of  the  observer  is  made  up  of  three 
parts:  those  due  to  the  motion  of  the  solar  system  as  a 
whole,  to  the  annual  motion  of  the  earth  in  its  orbit,  and 
to  the  diurnal  rotation  of  the  earth.     The  first  need  not 
be  considered,  since  it  affects  the  apparent  place  of  a  star 
by  a  constant  quantity.     The  second  gives  rise  to  annual 
aberration,  which  will  be  referred   to  in  CHAPTER  IV. 
The  third  gives  rise  to  the  diurnal  aberration.      This  is 
a  function   of   the   observer's  position  on  the  earth,  and 
will  be  treated  as  a  correction  to  be  applied  to  observed 
coordinates. 

38.  To  find  the  diurnal  aberration  in  hour  angle  and  dec- 
lination. 

In  Fig.  8  let  SENW^Q  the  horizon,  EQWi\&  equator, 
L  the  earth,  0  a  star  whose 
hour  angle  is  t  and  declina- 
tion 8,  EOW  &  great  circle 
through  0  and  the  east  point 
of   the   horizon.      Owing   to   w\ 
the   diurnal   rotation   of   the 
earth  the  observer  is  moving 
directly  toward  the  east  point, 
and  therefore  the   star's  ap- 
parent place  is  shifted  east-  FIG.  8 
ward  in  the  plane  UOWto  some  point  0'.    The  aberration 
in  this  plane  is  00',  whose  value  is  given  by  (108).     It 


42  PRACTICAL  ASTRONOMY 

only  remains  to  find  the  corresponding  change  in  hour 
angle  CO',  and  in  declination  CO  -  C'O'. 
In  the  triangle  EGO  we  have 

CO  =  8,        CE  =  90°  +  t,        ECO  =  90°. 
Now  let 

OE  =  y,        CEO^u,        C'C  =  dt,        C0-C'0f=d8, 

and  we  can  write 

sin  8  =       sin  y  sin  <u,  (109) 

sin  t  cos  B  =  —  cos  y,  (HO) 

cos  <  cos  8  =       sin  y  cos  a).  (HI) 

(110)  and  (111)  give  by  differentiation,  t,  S  and  7  being 
variables, 

—  sin  t  sin  8  c?8  -f  cos  £  cos  8  eft  =  sin  y  efy, 

—  cos  Z  sin  8  e?8  —  sin  £  cos  8  dt  =  cos  y  cos  o>  cfy. 

Eliminating  c?8  and  then  dt,  we  obtain 

cos  8  dt  =   (sin  y  cos  <  —  cos  y  sin  *  cos  <o)  efy, 
sin  8  dS  =  —  (sin  y  sin  t  +  cos  y  cos  <  cos  w)  efy, 

which,  by  means  of  (109),  (110)  and  (111),  reduce  to 


dt=     cos  t  sec  8-  (112) 

smy 

</8  =  -  sin  <  sin  8  -^--  (113) 

sin  y 

The  value  of  the  factor  -23L  is  given  by  (108)  from 

sin  7 

the  known  values  of  v  and  F".  For  an  observer  at  the 
earth's  equator  it  is  0".31  ;  in  latitude  <j>  it  is  0".31  cos</>. 
Substituting  this  value  in  (112)  and  (113)  we  obtain 

dt  =  +  0".31  cos  <#>  cos  t  sec  8,  (114) 

d8  =  -  0  .31  oos  <£  sin  t  sin  8,  (115) 

which  are  the  corrections  to  be  applied  to  the  observed 

hour  angle  and  declination. 

When  the  star  is  observed  on  the  meridian,  t  =  0,  and 

(114)  and  (115)  become 

<fr  =  0".31cos<£sec8,  (116) 

dS  =  0  .  (117) 


SEQUENCE   AND   DEGREE   OF    CORRECTIONS  43 

39.  To  find  the  diurnal  aberration  in  azimuth  and  alti* 
tude. 

The  problem  is  identical  with  that  in  §  38  save  that  the 
horizon  is  the  plane  of  reference,  instead  of  the  equator. 
If  in  (114)  and  (115)  we  replace  t  by  A  arid  B  by  h  we 
obtain  the  desired  corrections, 

dA  =  +  0".31  cos  <£  cos  A  sec  h,  (118) 

dh   -  _  0  .31  cos  <f>  sin  A  sin  /*,  (119) 

which  are  the  corrections  to  be  applied  to  the  observed 
azimuth  and  altitude. 

SEQUENCE   AND   DEGREE    OP   CORRECTIONS 

40.  In  applying  the  corrections  considered  in  this  chap- 
ter it  is  necessary  that  a  proper  sequence  be  followed. 

In  altitudes  measured  from  the  sea  horizon,  the  correc- 
tion for  dip  (103)  is  applied  previous  to  the  correction  for 
refraction.  In  all  other  cases  the  correction  for  refraction 
must  be  applied  first,  its  amount  being  obtained  by  the 
methods  of  §  30  arid  §  31. 

Except  in  a  few  cases  the  diurnal  aberration  may  be 
neglected. 

Observations  on  the  sun  or  moon  refer  to  points  on  the 
limb.  They  must  be  reduced  to  the  center.  In  the  case 
of  the  moon  the  reduction  is  made  by  formulae  (106)  and 
(107);  of  the  sun,  by  (107). 

The  parallax  is  now  determined  by  the  methods  of 
§§  25-29.  It  is  wholly  inappreciable  for  the  stars. 

The  degree  of  refinement  to  which  these  corrections 
should  be  carried,  can  be  stated  only  in  a  general  way. 
Usually  it  is  sufficient  to  compute  the  corrections  to  one 
order  of  units  lower  than  that  to  \vhich  the  observations 
have  been  made.  -Thus,  in  reducing  an  observation  made 
with  a  sextant  reading  to  10",  the  corrections  should  be 
computed  to  the  nearest  second.  If  the  mean  of  a  large 
number  of  sextant  readings  is  employed,  it  is  advisable 
to  carry  the  corrections  to  tenths  of  a  second ;  and  simi- 
larly in  other  cases. 


CHAPTER   IV 

PRECESSION  —  NUTATION  —  ANNUAL  ABERRATION 
—  PROPER  MOTION 

41.  In  the  preceding  chapter  we  considered  the  correc- 
tions necessary  to  be  applied  to  observed  coordinates  in 
order  to  reduce  them  to  the  center  of  the  earth.  We 
shall  now  consider  the  corrections  which  must  be  applied 
to  the  apparent  geocentric  coordinates. 

While  the  relative  positions  of  the  fixed  stars  change 
very  slowly,  —  and  in  most  cases  no  change  at  all  has 
been  detected,  —  their  apparent  coordinates  are  continually 
varying.  These  variations  are  divided  into  two  general 
classes,  secular  and  periodic. 

Secular  variations  are  very  slow  and  nearly  regular 
changes  covering  long  periods  of  time ;  so  that  for  a  few 
years,  and  in  some  cases  for  centuries,  they  may  be  re- 
garded as  proportional  to  the  time. 

Periodic  variations  are  changes  which  pass  quickly  from 
one  extreme  value  to  another,  so  that  they  cannot  be  treated 
as  proportional  to  the  time  except  for  very  short  intervals. 

The  planes  of  the  ecliptic  and  equator  are  subject  to 
slow  motions,  which  give  rise  to  variations  in  the  obliquity 
of  the  ecliptic  and  in  the  positions  of  the  equinoxes.  The 
coordinates  of  the  stars  therefore  undergo  changes  which 
do  not  arise  from  the  motions  of  the  stars  themselves,  but 
from  a  shifting  of  the  planes  of  reference  and  the  origin 
of  coordinates.  The  forces  producing  these  changes  are 
variable,  and  while  the  variations  of  the  coordinates  are 
progressive,  they  are  not  uniform.  They  may  be  regarded 

44 


PRECESSION  45 

i 

v 

as  made  up  of  two  parts,  viz.:  a  secular  variation  called 
precession,  and  a  periodic  variation  called  nutation. 

Owing  to  annual  aberration  [see  §  37]  the  stars  are 
not  seen  in  their  true  positions,  but  are  apparently  dis- 
placed toward  that  point  of  the  sphere  which  the  earth 
is  approaching,  thus  giving  rise  to  periodic  variations  of 
their  apparent  coordinates. 

In  the  case  of  stars  having  proper  motions, —  that  is, 
apparent  individual  motions  due  to  motions  of  the  stars 
themselves,  and  to  the  motion  of  the  solar  system  in  space, 
—  their  positions  on  the  sphere  change,  and  give  rise  to 
secular  variations  of  the  coordinates. 

42.  In  order  that  we  may  define  the  positions  of  the 
ecliptic  and  equator  at  any  instant,  it  will  be  convenient 
to  adopt  the  positions  of  these  planes  at  some  epoch  as 
fixed  planes,  to  which  their  positions  at  any  other  instant 
may  be  referred.  Let  their  positions  at  the  beginning  of 
the  year  1800  be  adopted  as  the  mean  ecliptic  and  equator 
at  that  instant. 

The  true  equator  and  ecliptic  at  any  instant  are  the  real 
equator  and  ecliptic  at  that  instant.  Their  positions  are 
affected  by  precession  and  nutation. 

The  positions  of  the  mean  equator  and  ecliptic  at  any 
instant  are  the  positions  these  circles  would  occupy  at 
that  instant  if  they  were  affected  by  precession,  but  not  by 
nutation. 

The  mean  place  of  a  star  at  any  instant  is  its  position 
referred  to  the  mean  equator  and  ecliptic  of  that  instant. 
It  is  affected  by  precession  and  proper  motion. 

The  true  place  of  a  star  is'  its  position  referred  to  the 
true  equator  and  ecliptic.  It  is  the  mean  place  plus  the 
variation  due  to  nutation. 

The  apparent  place  of  a  star  is  the  position  in  which  it 
would  be  seen  by  an  observer  (at  the  center  of  the  earth).  It 
is  the  true  place  plus  the  variation  due  to  annual  aberration. 


46  PRACTICAL   ASTRONOMY 

43.  In  solving  the  problems  considered  in  the  following 
chapters  we  require  to  know  the  apparent  right  ascensions 
and  declinations  of  the   celestial  objects  at  the  instants 
when  they  are  observed.     The  apparent  places  of  the  sun, 
moon,  major  planets,  and  several  hundred  of  the  brighter 
stars,  are  given  in  the  Ephemeris  at  intervals  such  that 
their  places  for  any  instant  may  be  obtained  by  interpola- 
tion.    But  occasionally  it  is  desirable  to  employ  stars  not 
included  in  this  list.     If  the  mean  places  of  these  stars  are 
given  in  the  Ephemeris  for  the  beginning  of  the  year* 
they  must  be  reduced,  by  means  of  the  proper  formulae,  to 
the  apparent  places  at  the  times  of  observation.     If  we 
observe  stars  which  are  not  contained  in  the  Ephemeris 
we    must   refer   for   their   positions  to  the   general   Star 
Catalogues,  which  contain  their  mean  places  for  the  begin- 
ning of  a  certain  year.     These  must  be  reduced  to  the 
corresponding  mean  places  for  the  beginning  of  the  year 
in  which  the  observations  are  made,  and  thence  to  the 
apparent   places   as   before.     We   shall   now  very  briefly 
consider  the  matters  essential  to  these  reductions. 

PRECESSION 

44.  If  from  the  figure  of  the  earth  we  subtract  a  sphere 
whose  radius  is  equal  to  the  earth's  polar  radius,  there  will 
remain  a  shell  of  matter  symmetrically  situated  with  refer- 
ence to  the  equator.     The  attractions  of  the  sun  and  moon 
on  this  shell  tend  to  draw  it  into  coincidence  with  the 
ecliptic.     This  tendency  is  resisted  by  the  diurnal  rotation 
of  the  earth.     The  combined  effect  of  these  forces  is  to 
shift   the    plane   of    the    equator,    without   changing   the 
obliquity  of  the  ecliptic,  in  such  a  way  that  its  intersec- 
tion with  the  ecliptic  continually  moves  to  the  west.     This 
causes  a  common  annual  increase  in  the  longitudes  of  the 

*  This  does  not  refer  to  the  ordinary  or  tropical  year,  but  to  the  fictitious 
year,  which  begins  at  the  instant  when  the  sun's  mean  longitude  is  280°. 


PRECESSION  47 

stars,  which  is  called  the  luni-solar  precession.  It  affects 
the  longitudes,  right  ascensions  and  declinations,  but  not 
the  latitudes. 

The  attractions  of  the  other  planets  upon  the  earth  tend 
to  draw  it  out  of  the  plane  in  which  it  is  revolving  around 
the  sun.  The  effect  is  to  shift  the  plane  of  the  ecliptic  in 
such  a  way  that  its  intersection  with  the  equator  moves  to 
the  east.  This  causes  a  small  annual  decrease  of  the  right 
ascensions  of  the  stars,  called  the  planetary  precession.  It 
affects  the  longitudes,  latitudes  and  right  ascensions,  but 
not  the  declinations. 

The  attractions  of  the  planets  produce  a  slight  change 
in  the  obliquity  of  the  ecliptic.  Its  annual  effect  upon  the 
coordinates  of  the  stars  is  combined  with  the  luni-solar  and 
planetary  precession,  the  whole  being  called  the  general 
precession. 

45.  These  motions  are  illustrated  in  Fig.  9.  Let  CVQ 
be  the  fixed  or  mean  ecliptic  at  the  beginning  of  the  year 
1800,  UV0  the  mean  equa- 
tor, and  VQ  the  mean  equi- 
nox. By  the  action  of  the 
sun  and  moon  in  the  time  t 
the  equator  is  shifted  to  the 
position  QVV  the  vernal 
equinox  moves  from  VQ  to 
Vv  and  VQVl  is  the  luni- 
solar  precession  in  the  in- 
terval t.  By  the  attraction 
of  the  planets  the  ecliptic  is  V3  -  V 

shifted  to  the  position  CVZ,  FlG*  9 

the  vernal  equinox  moves  from  Vl  to  F",  and  V-^  V  is  the 
planetary  precession  in  the  interval  t.  Let 

e0  =  the  mean  obliquity  of  the  ecliptic  for  1800  =CV0U, 
Ci  =  the  obliquity  of  the  fixed  ecliptic  for  1800  +  t  =  CV1Qt 
€  =  the  mean  obliquity  of  the  ecliptic  for  1800  +  t  =  CVQ, 


48  .  PRACTICAL   ASTRONOMY 

\l>  =  the  luni-solar  precession  in  the  interval  t  =  V0VV 
ft  =  the  planetary  precession  in  the  interval  t  =  V1VJ 
^=  the  general  precession  in  the  interval  t  =  CV  —  CV0. 

The  values  of  these  quantities  are,  according  to  Struve 
and  Peters,  referred  to  the  beginning  of  the  year  1800, 

c0  =  23°  27'  54".22,  1 

Cl  =  e0  +  0".00000735  t2, 

€  -  eo  _  0".4738  t  -  0".0000014  t2,  .19()x 

$  =  50".3798  t  -  0".0001084  t2, 

ft  =  0".15119  t  -  0".00024186  t2, 

^1=  50".2411  t  +  0".0001134  t2. 

46.  Given  the  mean  right  ascension  and  declination  (a,  £) 
of  a  star  for  any  date  1800  +  £,  required  the  mean  right 
ascension  and  declination  (a',  8')  for  any  other  date 
1800  + 1'. 

In  Fig.  9  let  OF2  be  the  ecliptic  of  1800,  V^Q  the 
mean  equator  of  1800 +  £,  arid  V^Q  the  mean  equator  of 
1800  + 1'.  If  we  distinguish  by  accents  the  values  given 
by  (120)  for  the  time  tr  we  have 


Now  let 

QFj  =  900-z,  QF2=90°  +  s',         VlQVt=$, 

and  we  have  [Chauvenef  s  Sph.  Trig.,  §  27] 

cos  \B  sin  \  (z1  +  2)  =  sin  J  (»//  -  ^)  cos  J  (c/  +  ej, 

COS  £  0  COS  j  (Z'  +  2)  =  COS  }  (^'  -  t/0  COS  i  (€/  -  Cj), 

sin"^  ^  sin  ^  (z'  -  z)  =  cos  \  ($'  -  «/0  sin  \  (c/  -  Cj), 
sin  $  0  cos  £  (z  —  2)  =  sin  ^  (^'  -  ^)  sin  4  (e/  +  ej. 

But  J  (z1  —  25)  and  J  (e/  —  et)  are  very  small  arcs,  and  we 
can  write 

tan  |  (zr  +  2)  =  tan  |  (<//  -  </0  cos  £  (e/  +  ex),  (121) 


tan  4  (f  -  «/0  sin  4  (e/  +  Cl) 
sin  }  0  =  sin  4  (^'  -  ^)  sin  4  (c/  +  Cl),  (123) 

which  determine  2f,  z  and  0  very  accurately. 


PRECESSION  49 

V  and  V  are  the  positions  of  the  mean  equinox  for 
1800  +  t  and  1800  +  tr.  Representing  the  planetary  pre- 
cessions V^V  and  F2F'  by  #  and  #',  we  have 

VQ  =  90°  -  z  -  #,  V'Q  =  90°  -f  z'  -  #'  ; 

and  since  for  the  star  $  we  have  a  =  F3f  and  af  =  V1  M'  , 
we  obtain 

MQ  =  90°  -  z  -  #  -  a,         M'Q  -  90°  +  2'  -  #'  -  a'. 


Then  if  P  and  P'  are  the  poles  of  the  mean  equator  at 
1800  +  t  and  1800  +  *',  and  if  we  put 

A=a  +  #  +  z,        A'  =  a!  +  #  -  z',  (124) 

we  have,  in  the  triangle  SPP', 

PS  =  90°  -  8,      P'S  =  90°  -  8',      PP1  =  FjQFi  =  0, 
SPP'  =  90°  -  Jf  Q  =  4,       ^P'P  =  90°  -f  M'Q  =  180°  -  A'. 

Substituting  these  in  (2)  and  (3)  we  obtain 

cos  8'  cos  A'  =  cos  8  cos  A  cos0  —  sin  8  sin0, 
cos  8'  sin  ,4'  =  cos  8  sin  A, 

from  which  wk  deduce 

cos  8'  sin  (A  '  -  A  )  =  cos  8  sin  A  sin  0  (tan  8  -f  tan  £  0  cos  A  )  ,  (125) 

sin  0  (tan  8  +  tan  %0  cos  4);  (126) 


or,  putting 

p  =  sin  0  (tan  8  +  tan£0cos4),  (127) 

we  have 


1-pcosA 

From  the  triangle  SPP'  we  can  also  obtain  [  Ohauvenet's 
Sph.  Trig.,  §  22] 

tan  J  (8'  -  8)  =  tan  $6  GOSK-4'  +  ^).  (129) 

"  ' 


Having  determined  ex,  i/r,  #,  ex',  ^  and  ft1  from  (120), 
z,  z'  and  0  from  (121),  (122)  and  (123),  and  A  from  (124), 
we  obtain  a1  from  (127),  (128)  and  (124),  and  8'  from 
(129). 

E 


50 


PRACTICAL   ASTRONOMY 


Example.     The  mean  place  of  Polaris  for  1755.0  was 
a  =  0*  43"'  42M1,         8  =  +  87°  59'  41".ll ; 

neglecting  proper  motion  what  will  be  its  mean  place  for 
1900.0? 

In  this  case  t  =  —  45  and  tf  =  +  100,  and  we  find,  from 

(120), 

e,  23°  27'  54".23488  q'        23°  27'  54".29350 

^  -  37   47  .31  i//  +    1   23  56  .90 

'  &           -     7  .29  tf'               +  12  .70 

and  therefore 


$  (€/  +  C^ 

23°  27'  54".26 

l(f-V')       + 

1      0  52  .10 
+    0  .02931 

tan  £  (\j/  -  ^)           8.248163 

sin  A         9.3125989 

ec»K«i'  +  «i)          9.962513 
|(^  +  «)        0°55'50".14 
log  i  (e/  -  Cl)           8.467016 
cot  \  (^'  -  ^)          1.751837 

logp         9.6050849 
cos  ,4          9.9906400 
log;?  cos  .4          9.5957249 
Sub*         0.2176761 

coseci  (£/  +  «!)          0.399910 
log  i  (2'  _  2)          0.618763 
i(2'-2)                    4".16 

tan  (A1  -  A)         9.1353599 
A'  -A        7°46'36".67 
A'      19  37  47  .01 

2'        0°  55'  54".30 

a!  =  A'  +  z'  -  tf'      20  33  28  .61 

2       0  55  45  .98 

a'        I*22m13«.91 

sinKf-tA)          8.248095 
sin  \  (e/  +  e^           9/600090 

A'  +  A      31°  28'  57".35 

^  ^       0°^24'  14".16 

tan£0         7.8481943 

a      10  55  31  .65 

cos$(A'  +  A)         9.9833991 

A  =  a  +  2  +  tf      11  51  10  .34 
tan  i(9          7.848194 

seei(4'-<4)        0.0010009 
tan  £(S'  -8)         7.8325943 

cos  A          9.990640 
tan  \0  cos  A          7.838834 
tan  8          1.4557773 

\  (8'  -  8)        0°  23'  22".85 
8'  -  8        0  46  45  .70 
8'      88  46  26  .81 

Add*          0.0001049 

sin  B          8.1492027 

log  jo          9.6050849 

47.    Required  the  annual  precession  in  right  ascension  and 
declination  at  any  time  1800  +  t. 

*  ZecWs  Tafeln  der  Additions-  und  Subtr actions-Log arithmcn  are  used 
here. 


PRECESSION  51 

The  precession  for  one  year  being  small  we  can  put,  in 
(124)  and  (125),  without  sensible  error, 

8'  =  8,     sin  (A'  -  A}  =  (A'  -  A)  sin  1",     sin  A  =  sin  a, 
sin  0  tan  ±0  =  0,     sin  0  =  0  sin  1", 

and  obtain 

A'  -A  =  a' -  a +  (#'-#)- (z' +  2)  =  0  sin  a  tan  8.         (130) 
For  (121)  and  (123)  we  may  write 

z'  -}-  z  =  (if/'  —  \l/)  cos  €v 
6  =(\l/'  —  i//)  sin  er 

Substituting  these  in  (130)  and  dividing  by  t1  —  t,  we  obtain 


l- 

e  -  1      tr  -t        1  .    t1  -t      f  -  1 
Similarly,  from  (129)  we  can  obtain 


sin  c,  cos  a. 


t'  -  t        t'  -t 

In  order  to  express  the  rate  of  change  in  a  and  B  at  the 
instant  1800  +  t  we  must  let  t'  —  t  become  very  small. 
Passing  to  the  limit  we  have 


dt       dt  dt       dt 

d$      d\l/   . 

—  =  —£  sin  c,  cos  a. 

dt       dt 

If  we  let  dt  equal  one  year,  and  put 


m=fcos£'-f'  n=fsint 


we  obtain  for  the  annual  precession  at  1800  + 1 

da. 
/\JL^Jy^A^n  ^v        —  =  ro  +  n  sin  a tan 8,  (131) 

*?  =  ncoso.  (132) 

tUQ  V        b-  rfl 
From  (120)  we  find 

^       ^L  cos  Cl  =  (50".3798  -  0".0002168  <)  cos  Cj 
=    46" .2135  -  0".0001989 1, 

d±=     0".1512-0".0004837/; 
dt 

:(AA    *  /*^<     fa  y  r- -4     "PjL     d  ^   ^    AW 

I  r 


52 


PRACTICAL   ASTRONOMY 


and  therefore 


m  =  46".0623  +  0".0002849  t, 
n  =  20".0607  -  0".0000863 1. 


(133) 
(134) 


Except  for  stars  near  the  poles  and  for  long  intervals  of 
time,  formulae  (131)  and  (132)  are  very  convenient  for 
computing  the  whole  precession  between  two  dates.  Thus 
if  it  is  required  to  determine  the  precession  in  a  and  8  from 
1800  -f- 1  to  1800  +  t',  we  first  obtain  approximate  values  of 
d  and  B  for  the  middle  date  1800  + 1  (t  +  t').  Using  these 
values  we  then  compute  the  annual  precession  for  this 
date,  which  is  approximately  the  average  annual  precession 
for  the  interval  t'  —  t,  and  thence  the  whole  precession  by 
multiplying  this  by  t'  —  t. 

It  is  convenient  to  have  the  values  of  m  and  n  given  by 
(133)  and  (134)  tabulated  as  follows : 


Date 

&* 

log  &  n 

log  w 

1750 

3s.  06987 

0.126348 

1.302439 

1760 

3  .07006 

0.126330 

1.302421 

1770 

3  .07025 

(1.126311 

1.302402 

1780 

3  .07044 

0.126292 

1.302383 

1790 

3  .07063 

0.126274 

1.302365 

1800 

3  .07082 

0.126255 

1.302346 

1810 

3  .07101 

0.126236 

1.302327 

1820 

3  .07120 

0.126218 

1.302309 

1830 

3  .07139 

0.126199 

1.302290 

1840 

3  .07158 

0.126180 

1.302271 

1850 

3  .07177 

0.126162 

1.302253 

1860 

3  .07196 

0.126143 

1.302234 

1870 

3  .07215 

0.126124 

1.302215 

1880 

3  .07234 

0.126106 

1.302197 

1890 

3  .07253 

0.126087 

1.302178 

1900 

3  .07272 

0.126068 

1.302159 

1910 

3  .07291 

0.126050 

1.302141 

1920 

3  .07310 

0.126031 

1.302122 

1930 

3  .07329 

0.126012 

1.302103 

1940 

3  .07348 

0.125994 

1.302085 

PRECESSION 


53 


Example.     The  mean  place  of  /3  Oridnis  for  1850.0  was 
a  =  5*  7m  19-.856  8  =  -  8°  22'  44".74 ; 

neglecting  proper  motion,  find  its  mean  place  for  1900.0. 

Using  the  values  of  m  and  n  for  the  middle  date  1875.0, 
and  a  and  B  for  1850.0,  we  may  obtain  very  nearly  the  an- 
nual precession  in  a  and  S  for  this  interval,  from  (131)  arid 
(132). 

0  .126115 
9  .988429 
9  .168186n 


log  &  n 

sin  a 

tan  8 

log      9  .282730, 
number  —  0*.  19175 

TV  m      3  .07224 

4*     2  .88049 
dt 


logn 
cos  a 


1.302206 
9.357543 


—      4".568 
dt 


The  approximate  coordinates  of  the  star  for  1875.0  are 
therefore 


a  =  5*  8"*  31«.87, 

Using  these  values  we  have 


8=  -8°20'50".5. 


log  T5  n 

sin  a 

tan  8 

losr 


0  .126115 
9  .988955 
9  .166514n 
9  .281584« 


number  -  08.19124 
&  m      3  .07224 

—      2  .88100 
dt 


logn 

cos  a 

rfS 

dt 


1.302206 
9.347705 

4".46592 


These  are  very  nearly  the  exact  values  of  the  annual  pre- 
cession for  1875.0,  and  the  mean  place  for  1900.0  is  there- 
fore 

a'  =  5»  9m  43'.906,  8'  =  -  8°  19'  1".44, 

which  is  practically  identical  with  that  given  by  the  rigor- 
ous method  of  §  46. 

,  In  many  star  catalogues  the  annual  precession  in  a  and 
8  is  given  for  each  star  for  the  epoch  of  the  catalogue,  by 
means  of  which  the  approximate  place  of  the  star  for  the 


54  PRACTICAL   ASTRONOMY 

middle  time  is  found  at  once,  and  the  first  approximation 
made  above  is  avoided. 

PROPER    MOTION 

48.  The  proper  motion  of  a  star  has  already  been  defined 
to  be  an  apparent  motion  of  the  star  itself  on  the  surface 
of  the  sphere.     It  is  assumed  to  take  place  in  the  arc  of  a 
great  circle,  and  to  be  uniform.     The  proper  motions  in 
right  ascension  and  declination  are  the  components  of  this 
motion  in  and  perpendicular  to  the  equator.     They  are 
variable  since  the  equator  is  a  moving  circle,  and  it  must 
be  specified  to  which  equator  they  refer. 

When  a  star's  place  is  required  to  be  known  very  accu- 
rately, its  position  should  be  taken  from  as  many  catalogues 
as  possible.  In  order  that  the  data  thus  obtained  may  be 
properly  combined,  a  thorough  knowledge  of  the  subject 
of  proper  motion  is  essential. 

49.  Given  the  observed  mean  places  (a,  S)  of  a  star  for 
1800  4-  1  and  («',  8')  for  1800  4-  «',  required  the  annual 
proper  motion. 

Starting  from  the  first  observed  place  and  computing  the 
precession  for  the  interval  tr  —  t  by  the  methods  of  §  46 
or  §  47,  let  the  resulting  place  for  1800  4-  1'  be  av  Sr  The 
discrepancies  «'  —  al  and  &'  —  Sx  are  due  to  proper  motion, 
and  the  annual  proper  motion  for  the  interval  is 

d*  =  y=4>  (135) 

referred  to  the  equator  of  1800  +  t'. 

Starting  from  the  second  observed  place,  computing  the 
precession  for  the  interval  t  —  t  ',  and  applying  it  to  ar  and 
8',  let  the  resulting  place  for  1800  4-  1  be  «2,  82.  The 
annual  proper  motion  for  the  interval  is 


(136> 

referred  to  the  equator  of  1800  +  t. 


PROPER   MOTION  55 

Example.  The  mean  places  of  Polaris  for  1755.0  and 
1900.0  given  in  NewcornVs  Standard  Stars  are 

for  1755.0,       a  =  0*  43™  42M1,       8  =4-87°  59'  41".ll, 
for  1900.0,       a!  =  1  22    33  .76,       8'  =  +  88  46  26  .66; 

determine  the  proper  motion  referred  to  the  equator  of 
1900.0. 

By  applying  the  precession  to  the  place  for  1755.0  the 
place  for  1900.0  was  found  to  be,  §  46, 

aL  =  1*  22™  13-.91,         Bl  =  +  88°  46'  26".81, 

and   therefore,  by   (135),  the   annual   proper   motion  of 
Polaris  referred  to  the  equator  of  1900.0  is 

da!  =  +  0-.1369,        do'  =  -  0".00103. 

50.  Griven  the  proper  motion  (c?a,  dB)  referred  to  the 
equator  of  1800  +  £,  required  the  corresponding  proper 
motion  (c?a',  dS1)  referred  to  the  equator  of  1800  +  tf, 
and  vice  versa. 

When  the  star  S  (Fig.  9)  moves  on  the  surface  of  the 
sphere  it  causes  variations  in  all  the  parts  of  the  triangle 
SP'P,  except  P'P.  The  solution  of  the  present  problem 
requires  a  knowledge  of  the  relations  existing  between 
these  variations. 

If  in  a  spherical  triangle  ABC  we  suppose  all  the  parts 
except  a  to  vary,  we  can  write  [Chauvenet's  Sph.  Trig., 
§  153,  (286)  and  (287)] 

sin  c  dE  =  sin  A  db  —  sin  a  cos  A  sin  B  cosec  A  dCt 
dc  =  cos  A  db  +  sin  a  sin  B  dC 

Substituting  in  these,  from  §  46, 

a  =  ppf  =  0,    db  =  d(SP)  =  -  d8,  dc  =  d(SP')  =  -  do', 
dE  =  d(SP'P)  =  rf(180°  -  A')  =  -  da!,  dC  =  d(SPP')  =  dA  =  da, 
and  putting  7  for  A,  we  obtain 

cos  8'  da!  =  sin  y  do  +  cos  8  cos  y  da,  (137) 

do'  =  cos  ydS  -  cos  8  sin  yda,  (138) 


56  PRACTICAL   ASTRONOMY 

in  which  7  is  determined  by 

sin  y  =  sin  0  sin  A  sec  8'  =  sin  0  sin  A '  sec  5,  (139) 

cos  y  =  (cos  0  —  sin  8  sin  8')  sec  8  sec  8'.  (140) 

These  determine  the  proper  motion  for  1800  -h  tr  in  terms 
of  that  for  1800  +  t. 

From  (137)  and  (138)  we  obtain 

cos  Bda  =  cos  8'  cos  y  da!  —  sin  y  d&',  (141 ) 

rf8  =  cos  8'  sin  y  cfa'  +  cos  y  d8',  (1^-) 

which  determine  the  proper  motion  for  1800  + 1  in  terms 
of  that  for  1800  +  *'. 

Example.  The  proper  motion  of  Polaris  referred  to  the 
equator  of  1900.0  is 

da!  =  +  OM369  =  -f  2".0535,        d&  =  -  0".00103. 

Deduce  the  proper  motion  referred  to  the  equator  of  1755.0. 
Using  0  and  A  from  §  46  and  $'  =  +  88°  46'  26".66  we 
find,  from  (139)  and  (140), 

sin  y  =  9.131493,        cos  y  =  9.995984, 
and  therefore  from  (141)  and  (142)  we  obtain 

da  =  +  1".2480  =  +  0«.0832,        dS  =  +  0".00493. 

51.  Given  the  proper  motion  (c?a,  dS~)  and  the  mean  place 
(a,  S)  of  a  star  for  the  epoch  1800  +  £,  required  its  mean 
place  (a',  $)  for  the  epoch  1800  +  tr. 

The  proper  motion  for  the  whole  interval  tr  —  t  is  first 
computed  and  applied  to  the  mean  place  for  1800  +  t. 
With  the  resulting  values  of  a  and  8,  which  we  shall  de- 
note by  ax  and  Sv  the  precession  is  computed  and  applied 
to  a1  and  8r  The  result  is  the  star's  mean  place  for 
1800  +  tr. 

If  the  proper  motion  (daf,  d$)  is  given  for  the  epoch 
1800  +  £',  we  first  compute  the  precession,  using  a  and  8, 
and  then  apply  the  proper  motion  for  the  interval. 


PROPER   MOTION  57 

Example  1.  Given  the  mean  place  and  proper  motion  of 
Polaris  for  1755.0, 

,  8  =  +  87°  59'  41". 11, 

d8  =  +    0".00493, 

required  the  mean  place  for  1900.0. 
The  proper  motion  for  the  interval  is 

+  (K0832  x  145  =  +  12-.06,        +  0".00493  x  145  =  -f  0".71. 

Therefore 

04  =  0*  43™  54M7,        ^  =  +  87°  59'  41".82. 

Employing  these  values  in  the  example  of  §  46  we  find 
for  1900.0 

a'  =  1*  22™  33«.76,         8'  =  -f  88°  46'  26".66. 

Example  2.  The  proper  motion  of  /3  Orionis  referred 
to  the  equator  of  1900.0  is 

da  =  -  0-.00027,        dS  =  -  0".0061. 

Include  this  in  the  example  of  §  47. 
The  proper  motion  for  the  interval  is 

-  (K00027  x  50  =  -  (K013,         -  0".0061  x  50  =  -  0".30; 
and  therefore  the  mean  place  for  1900.0  is 

a'  =  5»  9»  43«.893,         8'  =  -  8°  19'  1".74. 

52.  It  will  be  seen  from  §  47  that  the  annual  precession 
is  a  slowly  varying  quantity.  The  change  in  its  value  in 
one  hundred  years  is  called  the  secular  variation  of  the 
precession.  Many  star  catalogues  give  not  only  the  mean 
place  and  annual  precession  of  a  star  but  also  the  secular 
variation  and  proper  motion.  In  this  case  the  reduction 
of  the  mean  place  of  a  star  from  the  epoch  of  the  catalogue 
1800  +  t  to  that  for  1800  +  tr  is  readily  made.  For  if 

p  =  the  annual  precession  for  the  epoch  1800  -f  t, 
Ap  =  the  secular  variation, 
fi  =  the  proper  motion, 


58  PRACTICAL   ASTRONOMY 

the  reduction  for  the  interval  tf  —  t  will  be  the  annual 
change  for  the  middle  time  multiplied  by  t'  —  t,  or 

-*  (148) 


which  form  applies  both  to  the  right  ascension  and  the 
declination. 

Example.  NewcomVs  Standard  Stars  gives  the  follow- 
ing data  for  /3  Orionis  for  the  epoch  1850.0  : 

a  =       5*  7-  19*  .856,          8  =  -  8°  22'  44".74, 
p  =  +  2'  .87999,  p'=  +  4".5687, 

Ap  =  +  0-  .00400,  Ap'  =  -  0".4109, 
^  =  -  0»  .00025,  jx'  =  -  0".0061  ; 

required  the  mean  place  for  1900.0. 

Substituting  these  values  in  (143)  we  obtain  the  reduc- 
tions for  the  interval, 

for  a,  +  144«.038,        for  8,  -f  222".99, 
and  the  mean  place  for  1900.0  is,  therefore, 

a'  =  5*  9»  43«.894,        8'  =  -  8°  19'  I".  75. 

53.  Many  of  the  problems  of  practical  astronomy  require 
that  the  star  places  be  determined  with  the  utmost  accu- 
racy. In  such  cases  the  observed  coordinates  of  a  star  — 
along  with  the  corresponding  epochs  of  observation  —  are 
taken  from  as  many  star  catalogues  as  possible,  and  com- 
bined by  the  method  of  least  squares  in  order  to  determine 
the  most  probable  values  of  the  star's  coordinates  and 
proper  motion  at  any  given  time. 

Suppose  the  star's  right  ascension  is  given  in  n  cata- 
logues for  the  epochs  of  observation  tv  t2,  •••  tn,  and  that 
the  most  probable  values  of  the  right  ascension  and  proper 
motion  are  required  for  the  epoch  t.  Apply  the  precession 
up  to  the  instant  t  to  each  of  the  catalogue  positions,  and 
let  the  results  be  av  a2,  •••  an.  Let  p  be  the  star's  proper 
motion  referred  to  the  equator  of  the  epoch  t,  and  let  a  be 


PROPER   MOTION  59 

its  right  ascension  at  that  instant.     Then  we  shall  have 
n  equations 


a  -  a2  +  p.  (t  - 


(144) 


from  which  to  determine  the  most  probable  values  of  ft 
and  a.  To  put  them  in  a  form  suitable  for  solution,  let 
ft0  and  a0  be  approximate  values  of  ft  and  a,  and  let  Aft 
arid  Aa  be  small  corrections  to  them,  so  that 

Ij.  =  to  -f  A/A,  (145) 

a  =  a0  +  Aa ;  (146) 

then  equations  (144)  take  the  form 

(t  -  ^  )  A/A  -  Aa  +  [ttl  +  (<  -  O  to  -  ao]  =  °> 

(*  -  O  A/A  -  Aa  +  [an+  (t  -  tn)  ^  -  a0]  =  0. 


(147) 


The  solution  of  these  equations  will  determine  the  most 
probable  values  of  Aft  and  Aa,  and  therefore  of  ft  and  a, 
provided  the  n  original  data  are  of  equal  weight.  If  they 
are  of  unequal  weight,  as  will  almost  always  be  the  case, 
the  equations  (147)  must  be  multiplied  by  the  proper 
factors  before  proceeding  to  their  solution. 

The  weights  to  be  assigned  to  the  data  from  different 
star  catalogues  depend  upon  many  factors.  The  instru- 
ments and  methods  of  observation  and  reduction  employed, 
the  skill  of  the  observers,  and  the  number  of  individual 
observations  upon  which  the  printed  results  depend,  must 
all  be  taken  into  account.  Familiarity  with  the  methods 
of  meridian  circle  work  and  considerable  experience  with 
star  catalogues  are  necessary  acquirements  for  assigning 
suitable  weights.  Tables  of  relative  weights  in  the  intro- 
ductions to  NewcomVs  Standard  Stars  and  Boss's  500 
Stars  will  serve  as  partial  guides. 


60 


PRACTICAL   ASTRONOMY 


Example.  It  is  required  to  determine  the  most  probable 
values  of  the  right  ascension  and  the  proper  motion  in  right 
ascension  of  B  Trianguli,  for  the  epoch  1900.0. 

Observations  of  this  star  are  contained  in  twenty  or 
more  well-known  catalogues.  We  shall  select  ten  of  them, 
as  below.  The  first  column  contains  the  name  of  the  cata- 
logue, the  second  the  epoch  of  observation,  the  third  the 
catalogue  right  ascensions  corrected  for  the  precession  up 
to  1900.0,  and  the  fourth  the  relative  weights.  Except 
for  small  errors  of  observation,  the  discrepancies  in  column 


Catalogue 

Epoch  of 
Obs'n 

a 

W't 

a  1900.0 

Auwers-Bradley 

1755.0 

2»  10"*  43».72 

2 

2»  10™  57-.03 

Lalande 

1794.9 

47.03 

1 

56.69 

Piazzi 

1812.9 

48.42 

1 

56  .43 

Abo 

1830.0 

50.43 

2 

56.86 

Edinburgh 

1842.9 

51.68 

2 

56  .93 

Pulkowa 

1855.0 

52.72 

4 

56.86 

Greenwich  N.  7  yr. 

1864.0 

53  .63 

4 

56.94 

"             9  yr. 

1872.0 

54.36 

4 

56.93 

"            10  yr. 

1880.0 

55.07 

4 

56.91 

Cincinnati 

1890.6 

55.95 

4 

56.82 

three  are  due  to  proper  motion.  Comparing  the  first  and 
last  observations,  we  find  for  an  approximate  value  of  the 
annual  proper  motion,  /*0  =  -f-  05.09 ;  and  therefore  for  an 
approximate  value  of  the  right  ascension  at  1900.0, 
a0  =  2*  10m  56S.80.  We  may  now  write  equations  (147), 
thus : 

145.0  A/A-  Aa-  0.05  =  0,  ] 

(148) 

9.4  A/A  -  Aa  ±  0.00  =  0.  J 

Multiplying  these  equations  by  the  square  roots  of  their 
respective  weights,  and  combining  them,  we  obtain 


REDUCTION  TO  APPARENT  PLACE          61 

+  94792  A^t  -  1283  Aa  -  82.07  =  0,*  1 
-    1283  A/x  +      28Aa+    0.29-0,    J 
and  thence 

A/u  =  +  0-.0019,  Aa  =  +  0'.077. 

Substituting  these  and  /*0  and  a0  in  (145)  and  (146),  we 
obtain  the  most  probable  proper  motion  and  right  ascen- 
sion for  1900.0, 

fji  =  +  0-.09  +  Os.0019  -  +  Os.0919, 

a  =  2A  10-  56-.80  +•  (K077  =  2   10W  56*.877. 

The  last  column  of  the  table  above  contains  the  indi- 
vidual right  ascensions  corrected  for  proper  motion.  The 
modern  observations  are  in  good  agreement. 

An  entirely  analogous  method  would  be  used  to  deter- 
mine the  most  probable  values  of  the  declination,  and  the 
proper  motion  in  declination. 

REDUCTION  TO  APPARENT  PLACE 

54.  The  mean  place  of  a  star  for  the  beginning  of  the 
required  year  having  been  obtained  by  any  of  the  above 
methods,  it  remains  to  determine  its  apparent  place  for 
any  given  instant.  Thus  if  we  desire  the  apparent  place 
for  a  time  T  from  the  beginning  of  the  year,  we  obtain  the 
mean  place  by  adding  the  precession  and  proper  motion 
for  the  interval  r,  then  the  true  place  by  adding  the  nuta- 
tion, and  finally  the  apparent  place  by  adding  the  annual 
aberration.  The  reduction  to  the  mean  place  could  be 
performed  as  before ;  we  could  determine  the  nutation  by 
evaluating  the  long  and  tedious  nutation  formulae,  the 
deduction  of  which  belongs  to  physical  astronomy;  and 
we  could  obtain  the  annual  aberration  from  equations 
deduced  by  methods  analogous  to  those  of  §  38.  But 
this  process  is  laborious,  and  the  general  equations  are 
never  used  except  in  a  few  highly  specialized  problems. 

*  Another  computer  may  not  exactly  reproduce  these  coefficients,  on 
account  of  neglected  decimals. 


62  PRACTICAL  ASTRONOMY 

By  judiciously  combining  the  terms  of  the  various  for- 
mulse  involved  in  the  reductions,  Bessel  was  able  to  pro- 
pose two  simple  and  closely  related  methods,  which  are 
now  in  common  use.  We  shall  consider  them  in  the 
following  section. 

55.  Griven  the  mean  place  (a,  £)  of  a  star  for  the  begin- 
ning of  the  year,  required  its  apparent  place  (V,  8')  for  any 
instant  T* 

(a)    The  reduction  is  made  by  the  formulae 

a!  =  a  +  T/X  +  ACL  +  Bb  +  Cc  +  Dd  +  &  E,  (150) 

8'  =  8  +  T/A'  +  Act  +  Bb'  +  Cc'  +  Ddf,  (151) 

in  which  T/A  and  T/A'  represent  the  proper  motion  and  Aa 
and  Aar  the  precession  in  the  interval  T;  Bb  +  ^  J?and 
Bb'  the  nutation,  and  Cc  +  Dd  and  Cc'  +  Dd1  the  annual 
aberration  at  the  instant  T.  A,  B,  (7,  D,  and  E  are  the 
Besselian  star-numbers.  They  are  functions  of  the  time. 
The  American  Ephemeris  gives  their  general  values  on 
p.  280,  and  tabulates  the  logarithms  of  A,  B,  C,  and  D  for 
every  day  of  the  year  on  pp.  281-284.  The  value  of  E  is 
given  in  the  same  place.  It  is  a  slowly  varying  quantity 
whose  value  never  exceeds  0".05,  and  it  can  generally  be 
neglected. 

a,  6,  <?,  d,  ar,  br,  cf,  and  df  are  Bessel* s  star-constants. 
They  are  functions  of  the  star's  place  and  the  obliquity  of 
the  ecliptic,  and  are  defined  by  the  equations 

a  =  ^  m  +  -jJj  n  sin  a  tan  8,  a'  =  n  cos  a, 

b  =  TV  cos  a  tan  8,  V  =  -  sin  a, 

c  =  T^  cos  a  sec  8,  c'  =  tan  e  cos  8  —  sin  a  sin  8, 

d  =  fa  sin  a  sec  8,  d'  =  cos  a  sin  8. 

In  some  star  catalogues  the  logarithms  of  the  star-con- 
stants are  given  for  each  star.  But  these  values  become 
obsolete  in  a  few  years,  and  must  be  computed  anew  from 
(152),  since  m,  n,  a,  £,  and  e  are  variable  quantities. 

*  See  §  43,  footnote,  and  the  American  Ephemeris,  p.  280. 


REDUCTION  TO  APPARENT  PLACE          63 

Example.  Required  the  apparent  place  of  38  Lyncis  for 
the  upper  transit  at  Ann  Arbor,  1891  March  16. 

From  the  Berliner  Jahrbuch,  p.  180,  star  135,  we  find  for 
1891.0, 

a  =  9*  12~  3-.671,  8  =  +  37°  15'  48".43, 

H  =         -  0 .0030,  //  =  -  0  .114. 

The  upper  transit  occurs  therefore  at  Washington  side- 
real time  9*  39™,  or  lh  53m  before  mean  midnight.  Taking 
the  values  of  log  .A,  etc.,  from  the  American  Ephemeris, 
p.  281,  for  this  instant,  and  the  values  of  log  «,  etc.,  from 
the  Jahrbuch,  p.  329,  star  135,  the  computation  is  con- 
veniently arranged  as  below. 

log  a   0.5744  log&   8.5764n          logc   8.7943n          logd   8.7485 

log  A  9.0221n          log  B  0.5804n          log  C  1.2722n          log  D  0.1239 
log  a'  1.1734n          log  V  9.8254n          logc'  8.7763n          log<f  9.6533n 

a  9»12»3'.671  8  +  37°  15'  48".43 

T/X  -  0.001  Tfi'  -    0  .02 

Aa  -  0.395  Aa'  +    1  .57 

Bb  +0 .143  Bbf  +    2  .55 

Cc  +1.165  CJ  +    1.  12 

Dd  +  0.075  Dtf  -    0  .60 

•&E  -  0.003 

a!  9   12  4 .655  S'  +  37   15  53  .05 

This  method  of  reduction  should  be  employed  when  the 
star-constants  are  given  in  the  catalogues  with  sufficient 
accuracy,  or  when  the  apparent  places  of  the  same  star  are 
required  for  several  dates. 

In  using  the  data  of  reduction  furnished  by  the  British 
and  French  annuals  and  catalogues,  the  computer  must  be 
careful  to  follow  their  formulae :  for  while  the  form  of 
reduction  usually  agrees  with  the  American  and  German 
form,  the  notation  is  different,  A  and  B  in  the  one  corre- 
sponding to  0  and  D  respectively  in  the  other.  This 
applies  also  to  the  American  Ephemeris  previous  to  1865. 

(ft)  When  the  catalogues  do  not  give  the  values  of  log 
a,  log  ft,  etc.,  and  when  only  one  or  two  places  of  the  same 


64 


PRACTICAL    ASTRONOMY 


star  are  desired,  another  form  of  reduction  is  preferable. 

If  we  put 

/=  A  m  A  +  &  E>  A  sin  //  =  C, 

g  sin  G  =  B,  h  cos  H  =  D, 

g  cos  G  =  n  A ,  i  =  C  tan  e, 

the  formulae  (150)  and  (151)  become 

a'  =  a  +  T/X  +/+  &g  sin  (<7  +  a)  tan  8  +  ^  A  sin  (H  +  a)  sec  8,  (153) 
8'  =  8  +  T/A'  +  <7  cos  (G  +  a)  +  /<  cos  (H  +  a)  sin  8  +  i  cos  8,  (154) 

in  which  the  terms  involving /,#  and  G-  denote  the  pre- 
cession and  nutation,  and  the  terms  involving  A,  If  and  i, 
the  annual  aberration.  These  auxiliary  quantities  are 
called  the  independent  star-numbers.  The  values  of  T,/, 
(r,  H,  log  #,  log  h  and  log  i  are  given  in  the  American 
Ephemeris,  pp.  285-292,  for  every  day  of  the  year. 

Example.  Required  the  apparent  place  of  38  Lyncis  for 
the  upper  transit  at  Ann  Arbor,  1891  March  16. 

Using  the  data  given  above,  the  computation  is  con- 
veniently made  as  below. 


G 

240°  59' 

l°g<7 

0.6387 

a 

9*  12m  3* 

.671 

a 

138     1 

cos(<7  +  a) 

9.9757 

T/X 

-    0 

.001 

H 

274     3 

f 

-    0 

.326 

log  h 

1.2733 

(1) 

+  o 

.072 

l°g  i1? 

8.8239 

cos  (H  +  a) 

9.7887 

(2) 

+     1 

.240 

log  g 

0.6387 

sin  8 

9.7821 

a' 

9  12    4 

.656 

sin((r  +  a) 

9.5126 

log  (4) 

0.8441 

tan  8 

9.8813 

8 

+  37°  15'  48" 

.43 

log(l) 

8.8565 

log; 

0.9096n 

V 

-    0 

.02 

cos  8 

9.9008 

(3) 

+    4 

.12 

log  A 

8.8239 

(4) 

+    6 

.98 

logfc 

1.2733 

(5) 

-    6 

.46 

sin  (H  +  a) 

9.8969 

a' 

+  37   1553 

.05 

sec  8 

0.0992 

. 

log  (2) 

0.0933 

CHAPTER  V 
ANGLE   AND   TIME   MEASUREMENT 

56.  The  degree  of  refinement  to  which  an  observer  can 
carry  the  determination  of  his  geographical  position  and 
the  time  depends  in  general  upon  the  accuracy  attainable 
in  pointing  the  telescope,  in  reading  the  angle  corresponding 
to  the  pointing,  and  in  noting  the  time  when  the  pointing 
is  made.     In  general,  these  elements  are  of  equal  impor- 
tance.   For  any  given  telescope  the  first  depends  upon  the 
observer's  skill.     In  the  last  two  the  observer's  skill  is 
assisted  by  various  mechanical  devices. 

THE   VERNIER 

57.  An  angle  is  usually  measured  by  means  of  a  gradu- 
ated circle,  or  arc,  whose  center  is  at  the  vertex  of  the 
angle.     Closely  fitting  upon  the  graduated  arc  of  the  cir- 
cle and  centered  with  it  is  another  graduated  arc  called  the 
vernier,  which  is  so  arranged  upon  an  arm  that  it  moves 
with  reference  to  the  circle  when  the  telescope  moves. 
The  angle  to  be  read  is  that  included  between  the  zero 
line  of  the  circle  and  the  zero  line  of  the  vernier.     The 
zero  of  the  vernier  generally  falls  between  two  consecutive 
lines  on  the  circle.     The  angle  corresponding  to  the  whole 
divisions  can  be  read  off  at  once :  it  is  the  object  of  the 
vernier  to  determine  the  fractional  part  of  a  division.     It 
is  so  constructed  that  n  of  its  divisions  are  equal  in  length 
to  7i  —  1  divisions  of  the  circle.     If  we  let 

d  =  the  value  of  one  division  of  the  circle, 
d'  =  the  value  of  one  division  of  the  vernier, 
r  65 


66 

we  have 


PRACTICAL   ASTRONOMY 


or 


(155) 


d  —  d'  is  called  the  least  reading  of  the  vernier.  If  now 
the  zero  of  the  vernier  coincides  with  a  division  line  of  the 
circle,  the  circle  reading  gives  the  required  angle  at  once. 
If  the  first  vernier  line  coincides  with  a  circle  line,  the  zero 
of  the  vernier  is  d  —  d'  beyond  a  line  of  the  circle,  and  the 
circle  reading  must  be  increased  by  the  least  reading.  If 
the  second  vernier  line  is  in  coincidence  with  a  circle  line, 
the  circle  reading  must  be  increased  by  twice  the  least 
reading,  etc.  For  example,  the  value  of  a  division  of  a 
sextant  is  10',  and  60  divisions  of  the  vernier  correspond 
in  length  to  59  divisions  of  the  circle.  The  least  reading 
is  10'  -j-  60  =  10".  In  measuring  a  certain  angle  the  zero 
of  the  vernier  fell  between  42°  40'  and  42°  50',  and  the 
26th  line  of  the  vernier  coincided  with  a  circle  line.  The 
required  reading  was  42°  40'  +  26  x  10"  =  42°  44'  20';. 
In  practice  no  computation  is  necessary,  the  number  of 
minutes  being  read  directly  from  the  numbers  on  the 
vernier. 

In  the  accompanying  illustration,  Fig.  10,  there  are  two 
verniers  on  the  upper  graduated  arc :  one  to  the  right  of 


A,  to  be  used  in  connection  with  the  inner  set  of  readings 
on  the  circle  numbered  from  10°  to  the  right;  and  one  to 


THE   READING   MICROSCOPE  67 

the  left  of  A,  to  be  used  in  connection  with  the  outer  set 
of  readings  numbered  from  140°  to  the  left.  From  (155) 
it  follows  that  the  least  reading  of  the  vernier  is  I/.  The 
circle  reading,  when  read  to  the  right,  is  27°  25' ;  and 
when  read  to  the  left,  152°  35'. 


THE   READING   MICROSCOPE 

58.  In  very  fine  instruments  the  vernier  is  replaced  by  a 
reading  microscope,  the  optical  axis  of  which  is  perpendicu- 
lar to  the  plane  of  the  graduated  circle.  The  microscope 
is  so  adjusted  that  an  image  of  the  circle  divisions  is 
formed  in  the  common  focus  of  the  objective  and  ocular. 
In  the  same  focus  are  two  very  fine  micrometer  wires 
(usually  spider-lines)  which  either  intersect  at  a  small 
angle,  or  are  parallel  and  close  together.  In  the  former 
case  they  are  adjusted  so  that  the  bisector  of  their  acute 
angle  is  parallel  to  the  image  of  the  circle  graduation  seen 
nearest  the  middle  of  the  field  of  view.  In  the  latter 
case,  the  wires  are  made  parallel  to  that  graduation.  They 
are  stretched  upon  a  light  frame  whose  plane  is  parallel  to 
the  plane  of  the  circle,  and  which  may  be  moved  in  that 
plane  in  the  direction  at  right  angles  to  the  visible  gradu- 
ations by  turning  a  fine  micrometer  screw.  Fixed  upon 
the  projecting  end  of  the  screw  is  a  cylindrical  micrometer 
head.  This  is  graduated  into  either  60  or  100  parts,  and 
is  used  for  reading  the  fractional  parts  of  a  revolution  of  the 
screw,  the  readings  being  made  with  reference  to  a  fixed 
index.  The  whole  number  of  revolutions  is  indicated 
by  a  scale  sometimes  inside,  and  again  outside,  of  the 
microscope. 

Let  the  micrometer  screw  be  turned  until  the  wires  are 
in  the  center  of  the  field  of  view  and  the  reading  of  the 
head  is  zero.  The  position  now  occupied  by  the  wires  is 
the  fixed  point  of  reference.  The  angle  to  be  read  is  that 
included  between  the  zero  of  the  circle  and  this  point.  If 


68  PRACTICAL    ASTRONOMY 

now  the  micrometer  wires  coincide  with  a  line  of  the  circle 
the  desired  reading  is  obtained  at  once.  If  they  fall  be- 
yond a  certain  line  the  fractional  part  of  a  division  is  deter- 
mined by  moving  the  wires  from  the  point  of  reference 
into  coincidence  with  the  line.  The  distance  passed  over 
is  determined  from  the  micrometer  reading  and  the  known 
angular  value  of  one  revolution  of  the  screw.  In  setting 
the  wires  upon  any  circle  division  the  last  motion  of  the 
micrometer  should  always  take  place  in  the  direction  which 
increases  the  tension  on  the  screw,  any  lost  or  dead  motion 
being  thereby  avoided. 

When  the  microscope  is  properly  adjusted,  a  whole 
number  of  revolutions  of  the  screw  corresponds  exactly  to 
the  distance  between  two  consecutive  circle  lines.  But 
this  adjustment  once  made  does  not  remain,  owing  to 
changes  of  temperature,  etc.  It  is  customary  to  determine 
from  time  to  time  the  error  which  arises  and  allow  for  it. 
This  is  called  the  error  of  runs.  Its  value  is  found  by 
measuring  several  divisions  in  different  parts  of  the  circle, 
and  taking  the  mean  of  the  measures  in  order  to  eliminate 
as  far  as  possible  any  errors  in  the  graduations.  To  illus- 
trate, let  a  circle  be  graduated  to  5',  let  the  value  of  a  revo- 
lution of  the  screw  be  1',  and  let  the  head  be  divided  into  60 
parts.  Let  the  mean  of  the  measures  of  ten  divisions  in 
different  parts  of  the  circle  be  4  revolutions  and  56.4  divi- 
sions of  the  head.  The  correction  for  runs  per  minute  of 
arc  is  -f  3".6  -^  5  =  -f  0".72.  Let  an  angle  be  read  such 
that  the  circle  graduation  employed  is  62°  15',  and  the 
micrometer  reading  is  2  rev.  15.9  div.  The  correction  for 
runs  is  +  1".6,  and  the  angle  is,  therefore,  62°  17'  17".5. 

Better  still,  the  program  of  observations  should  if  pos- 
sible include  microscope  readings  on  the  two  graduations 
nearest  the  middle  of  the  field  of  view,  instead  of  on  only 
one.  Each  complete  observation  would  then  furnish  the 
necessary  data  to  correct  for  error  of  runs.  This  excellent 
practice  is  illustrated  in  the  example  of  §  126. 


ECCENTRICITY 


69 


FIG.  11 


ECCENTRICITY 

59.  The  center  of  the  arm  which  carries  the  vernier  or 
microscope  never  coincides  ex- 
actly with  the  center  of  the 
circle,  and  an  error  due  to  this 
eccentricity  enters  into  the  circle 
readings.  In  Fig.  11,  let  C'  be 
the  center  of  the  vernier,  C  the 
center  of  the  circle,  D  the  point  A 
of  intersection  of  the  circle  DAB 
and  the  line  CO'  produced,  A 
the  zero  point  of  the  circle,  and 
M  the  position  of  the  vernier  or 
microscope.  The  pointing  of 
the  telescope  corresponds  to  the 
direction  C'M  while  the  circle  reading  refers  to  the  direc- 
tion CM.  The  correction  for  eccentricity  is  therefore 
C'MC.  To  find  its  value  let 

CC'  =  the  eccentricity  =  <?, 

c  =  the  correction  for  eccentricity  =  C'MC, 
M  =  the  'observed  reading  of  the  circle, 
R  =  the  true  reading  of  the  circle  =  M  +  e, 
r  =  the  radius  of  the  circle, 
r)  =  the  angle  DC' A, 
v  =  the  angle  A  C'M. 

From  the  triangle  C'MC  we  can  write 

r  sin  e  =  e  sin  (y  +  17). 
Since  r  is  the  unit  radius  and  e  is  very  small,  we  have 

e  =  -^—  sin  0  +  17)  =  e"  sin  (V  +  rj),  (156) 

sin  1 

and  the  true  reading  of  the  circle  is 

R  =  M+e"  sin  (v  +  77).  (157) 

The  vernier  arm  MC'  is  often  produced  to  the  oppo- 
site point  of  the  circle,  which  call  M\  and  carries  another 


70  PRACTICAL   ASTRONOMY 

vernier  or  microscope.  The  minutes  and  seconds  of  the 
circle  reading  at  this  point  being  obtained  (a  second  vernier 
is  not  necessary  to  determine  the  degrees),  if  M'  is  the 
observed  reading,  the  true  reading  R  is  given  by 

R  =  M'  +  e"  sin  (180°  +  v  +  77)  =  M1  -  e"  sin  (v  +  ry). 
Combining  this  with  (157)  we  have 

R  =  \(M+M')',  (158) 

from  which  it  appears  that  the  eccentricity  is  fully  elimi- 
nated by  taking  the  mean  of  two  readings  180°  apart.  It 
can  be  shown  also  that  it  is  eliminated  by  using  the  mean 
reading  of  any  number  of  equidistant  microscopes. 

THE  MICROMETER 

60.  If  the  angle  between  two  points  is  smaller  than  the 
angular  diameter  of  the  field  of  the  telescope,  it  is  most 
easily  and  accurately  measured  by  means  of  a  micrometer. 
This  is  the  same  in  principle  as  that  used  on  the  reading 
microscope,  save  that  the  movable  wire  is  usually  composed 
of  a  single  thread,  and  is  generally  accompanied  by  other 
wires.  There  is  usually  one  fixed  wire  parallel  to  the 
movable  wire,  and  often  at  least  one  transverse  fixed  wire 
perpendicular  to  it.  The  arrangement  of  the  wires  varies 
to  meet  the  requirements  of  different  problems  and  the 
preferences  of  the  observers.  The  plane  of  the  wires  is  in 
the  common  focal  plane  of  the  objective  and  eyepiece  of  the 
telescope. 

The  micrometer  is  so  constructed  that  it  can  be,  and 
always  is,  rotated  about  the  line  of  sight  until  the  microm- 
eter wire  is  perpendicular  to  the  plane  of  the  angle  which 
it  is  desired  to  measure.  In  the  transit  instrument  the 
movable  wire  of  the  micrometer  is  vertical,  and  in  the 
zenith  telescope  it  is  horizontal.  [See  §  110,  and  Fig.  20.] 
The  modern  meridian  circle  is  provided  with  both  hori- 
zontal and  vertical  micrometer  wires.  The  filar  microm- 


THE   MICROMETER 


71 


eter  of  an  equatorial  telescope  is  arranged  so  that  the  wires 
can  be  turned  into  any  position,  and  their  direction  may 
be  determined  by  means 
of  a  graduated  position 
circle. 

Figure  12  represents 
the  filar  micrometer  of  the 
12-inch  equatorial  of  the 
Lick  Observatory,  by  Al- 
van  Clark  &  Sons.  The 
wires  are  in  the  box  &. 
To  render  them  visible  at 
night,  they  are  illumi- 
nated by  the  lamp  L,  sup- 
ported by  the  framework 

JQRPV  (designed  by 
Burnham),  and  counter- 
balanced by  IHF.  The 

graduated  position  circle 

XY  remains   fixed  with 

reference  to  the  telescope, 

whereas   the    micrometer 

box  (and  the  illuminating 

apparatus)  may  be  ro- 
tated about  the  line  of 

sight  so  as  to  place  the 

wires   in    any   direction. 

Their   direction    will    be 

indicated    by   the    circle 

readings    at    X   and    Y. 

Further,  by  turning   the 

screw  EE'  the.  microme- 
ter box,  with  the  entire 

system   of  wires,  can  be 

moved    in  a  direction   parallel  to  the  micrometer  screw. 

The   system   of  wires   may   therefore   be   given   motions 


FIG.  12 


72  PRACTICAL   ASTRONOMY 

of  rotation  and  translation,  to  place  them  in  any  desired 
position. 

The  light  from  the  lamp  shines  in  the  direction  TO,  but 
at  the  intersection  of  the  tubes  TO  and  NM  there  is  a 
diagonal  mirror  which  reflects  the  light  in  the  direction 
NV  into  the  box.  The  mirror  may  be  rotated  by  rotating 
a  disk  at  0,  thereby  varying  the  intensity  of  the  illumina- 
tion. If  electric  lighting  is  available,  the  oil  lamp  L  should 
be  replaced  by  a  small  incandescent  lamp,  as  the  latter  has 
many  practical  advantages. 

The  distance  between  the  fixed  and  the  movable  wires, 
or  the  distance  between  two  positions  of  the  movable  wire, 
is  indicated  by  the  readings  of  the  graduate^  micrometer 
heads  A  and  B :  A  indicating  the  whole  number  of  revolu- 
tions of  the  screw  and  B  the  fractional  parts  of  a  revolu- 
tion. To  convert  the  readings  into  arc,  the  value  of  one 

revolution  of  the  screw  must  be  known. 

\ 

61.  The  angular  value  of  a  revolution  of  the  micrometer 
screw  depends  upon  the  pitch  of  the  screw  and  the  focal 
length  of  the  telescope.  It  may  be  found  in  several 
ways. 

(a)  By  the  methods  described  above,  measure  with  the 
micrometer  any  known  angle,  and  divide  the  number  of 
seconds  in  the  angle  by  the  corresponding  number  of  revo- 
lutions of  the  screw. 

If  the  distance  between  two  stars  is  measured,  the  true 
distance  must  be  corrected  for  refraction.  A  method  com- 
monly employed  consists  in  measuring  the  difference  of 
declination  of  two  selected  stars  in  the  Pleiades  when  that 
group  is  near  the  meridian.  The  positions  of  these  stars 
are  very  accurately  known,  pairs  of  almost  any  desired 
distance  can  be  selected,  and  the  correction  for  refraction 
is  simple. 

Example.  The  difference  of  declination  of  d  Ursce  Majo- 
ris  and  Groombridge  1564  was  measured  with  the  movable 


THE   MICROMETER 


73 


wire  of  the  micrometer  of  the  zenith  telescope  of  the 
Detroit  Observatory,  when  on  the  meridian,  1891  March 
28.  Barom.  29.206  inches,  Att.  Therm.  58°.0  F.,  Ext. 
Therm.  37°. 5  F.  Find  the  value  of  a  revolution  of  the 
screw.  The  zenith  level  was  read  immediately  after  bisect- 
ing each  star,  in  order  to  correct  for  any  change  in  the 
pointing  of  the  telescope.  The  value  of  one  division 
of  the  level  is  2".74. 


crj.n 

Level 

n 

s 

d  Ufsce  Maj. 

9*  25™ 

+  70°  18'  46".5 

40.4 

18.7 

48.566 

Gr.  1564. 

9   33 

+  69   44  12  .7 

41.3 

19.5 

2.598 

The  correction  for  level  (see  §  62)  is  0.85d=2".3,  by 
which  amount  the  measured  distance  must  be  increased,  or 
the  difference  of  the  declinations  decreased.  The  differ- 
ence of  the  refractions  for  the  two  stars  is  0''.7,  by  which 
amount  the  difference  of  the  declinations  must  be  decreased. 
The  corrected  difference  of  the  declinations  is  34'  30". 8. 
Therefore  the  value  of  one  revolution  of  the  screw  is 
45".05. 

(6)  A  more  accurate  value  is  obtained  by  observations 
on  one  of  the  close  circumpolar  stars.  The  telescope  is 
directed  so  that  the  star  is  just  entering  the  field,  and  will 
be  carried  through  the  center  by  its  diurnal  motion.  The 
micrometer  is  revolved  so  that  the  micrometer  wire  will  be 
perpendicular  to  the  diurnal  motion  of  the  star  when  it 
passes  through  the  center  of  the  field.  The  wire  is  set  just 
in  advance  of  the  star,  the  time  of  transit  of  the  star  over  it 
is  noted,  and  the  micrometer  is  read.  The  wire  is  moved 
forward  one  revolution,  or  a  part  of  a  revolution,  and  the 
transit  observed  as  before.  In  this  way  the  observations 
are  carried  nearly  across  the  field. 


74  PRACTICAL   ASTRONOMY 

In  Fig.  13,  let  P  be  the  pole,  EP  the  observer's  meridian, 
SS'  the  diurnal  path  of   a  star,  AS  the  position  of   the 
micrometer   wire  when   at  the  center 
of   the  field  and  coincident   with   an 
hour  circle  PM,  and  BS'  (parallel  to 
AS)  any  other  position   of  the  wire. 
Now  let  mQ  be  the  micrometer  reading, 
tQ  the  hour  angle,  and  T0  the  sidereal 
time  when  the  star  is  at  S,  and  let  m, 
t  and  T  be  the  corresponding  quanti- 
;^     .        ties  when  the  star  is  at  S',  and   let 
is  -R  be  the  value  of  one  revolution  of 

the  screw.  Through  S'  pass  an  arc 
of  a  great  circle  8'  G  perpendicular  to  AS.  Then  in  the 
triangle  CS'P,  right-angled  at  (7,  we  have 

CS'  =  (m  -  wz0)72,     S'P  =  90°  -  8,     CPS'  =  t  -  t0  =  T  -  !T0; 
and  we  can  write 

sin  [(m  -  w0)72]  =  sin  (  T  -  T0)  cos  B  ; 
or,  since  (m  —  m^R  is  always  a  small  angle, 

(m  -7110)72  =  sin<  T  -  !T0)  ^-|-  (159) 

S1H  x 

Similarly,  for  another  observation, 


Sill  JL 

Combining  these  to  eliminate  the  zero  point, 


sm(r-  T0)-,      (160) 
sin  l  sin  i. 

from  which  the  value  of  R  is  obtained.  The  micrometer 
readings  are  supposed  to  increase  with  the  time. 

The  times  of  transit  are  supposed  to  be  noted  by  means 
of  a  sidereal  time-piece.  If  its  rate  [§  64]  is  large  it  must 
be  allowed  for.  If  a  mean  time-piece  is  used  the  intervals 
T—  TQ  must  be  converted  into  sidereal  intervals. 

The  method  is  advantageous  for  a  meridian  instrument 
with  a  micrometer  in  right  ascension,  the  star  being 
observed  at  upper  or  lower  culmination. 


THE  MICROMETER  75 

The  value  of  E  given  by  (160)  requires  a  correction 
for  refraction.  [The  development  of  the  formula  for 
determining  the  refraction  correction  and  its  application 
to  the  problem  on  pp.  75-77  are  on  p.  260.]  Further, 
any  variations  in  the  azimuth  or  level  constants  of  the 
instrument  during  the  progress  of  the  observations  in- 
troduce errors  in  the  results.  If  a  and  b  are  the  values 
of  these  constants  at  the  beginning  of  the  series  of  tran- 
sits and  a'  and  b'  their  values  at  the  close  of  the  series, 
it  can  be  shown  from  the  theory  of  the  transit  instru- 
ment [Chapter  VII],  that  the  distance  between  the  first 
and  last  positions  of  the  wires  has  been  decreased  by 
the  quantity 

(a'  -  a)  sin  (<£  ^  8)  +  (V  -  b)  cos  (<£  =f  3),  (161) 

which  divided  by  the  corresponding  difference  of  the 
micrometer  readings  is  the  correction  to  the  value  of  one 
revolution  of  the  screw.  The  lower  signs  are  for  lower 
culmination. 

The  azimuth  constants  are  determined  by  observing  suit- 
able pairs  of  stars  before  and  after  the  series  of  micrometer 
transits  is  taken,  according  to  the  methods  described  later. 
The  level  constants  are  determined  by  the  method  of  §  62. 
The  variations  of  azimuth  and  level  may  be  considered  to 
be  uniform  and  proportional  to  the  time.  For  a  meridian 
instrument  properly  mounted  the  variation  of  the  azimuth 
may  be  neglected  without  a  sacrifice  of  accuracy. 

Example.  Polaris  was  observed  at  lower  culmination 
at  Ann  Arbor,  1891  March  28,  to  determine  the  value  of  a 
revolution  of  the  micrometer  screw  of  the  transit  instru- 
ment. The  micrometer  was  set  at  every  three-tenths  of  a 
revolution,  and  one  hundred  and  fifty  transits  observed. 
The  times  were  noted  by  means  of  a  sidereal  chronometer 
which  was  16m  30S.6  slow.  The  position  of  Polaris  was 

*  a  =  1*  17"  46-.0,          8  =  +  88°  43'  40".25, 


76 


PRACTICAL   ASTRONOMY 


American  Ephemeris,  p.  304 ;  and  therefore  the  chronom- 
eter time  of  lower  culmination  was  T9  =  13A  lm  15*.4.  A  few 
of  the  observations  and  their  reduction  are  given  below. 
Each  printed  observation  is  the  mean  of  three  consecutive 
original  observations. 


ra 
e«r 

T 

T-T0 

T-  T0 

sin  (T-  T0) 

(m  —  m0)  R 

-> 

12*  23"*  12«.3 

—  38"*  3«.l 

—  9°  30'  46".5 

9.218194W 

—  756".83 

8.7. 

25  17  .3 

35  58.1 

8  59  31  .5 

9.193953n 

715  .75 

9.6 

27  19.7 

33  55.7 

8  28  55  .5 

9.168793,, 

675  .46 

10.5 

29  22.0 

31  53  .4 

7  58  21  .0 

9.142070n 

635  .15 

11.4 

31  22  .0 

29  53  .4 

7  28  21  .0 

9.11411U 

595  .55 

20.4 

51  42.3 

9  33  .1 

2  23  16  .5 

8.61977  U 

190  .80 

21.3 

53  43  .3 

7  32  .1 

1  53  1  .5 

8.516822,, 

150  .53 

22.2 

55  46  .0 

5  29.4 

1  22  21  .0 

8.379348* 

109  .69 

23.1 

57  47  .0 

3  28.4 

0  52  6  .0 

8.180547n 

69  .40 

24.0 

59  50.0 

-  1  25  .4 

-0  21  21  .0 

7.793121,, 

-  28  .44 

25.8 

13  3  54  .0 

+  2  38  .6 

+  0  39  39  .0 

8.061960 

+  52  .82 

2(5.7 

5  ,58  .3 

4  42  .9 

1  10  43  .5 

8.313268 

94  .21 

27.6 

7  59  .5 

6  44.1 

1  41  1  .5 

8.468092 

134  .55 

28.5 

10  0  .0 

8  44.6 

2  11  9  .0 

8.581389 

174  .66 

29.4 

12   3.7 

10  48  3 

4  42  4  .5 

8.673281 

215  .82 

38.4 

32  24.3 

31   8.9 

7  47  13  .5 

9.131914 

620  .48 

39.3 

34  28  .3 

33  12  .9 

8  18  13  .5 

9.159630 

661  .20 

40.2' 

36  32  .2 

35  16  .8 

8  49  12  .0 

9.185629 

702  .16 

41.1 

38  34  .0 

37  18  .6  , 

9  19  39  .0 

9.209722 

742  .21 

42.0 

13  40  37  .0 

+  39  21  .6 

+  9  50  24  .0 

9.232735 

+  782  .60 

Subtracting  the  1st  from  the  llth,  the  2d  from  the  12th, 
etc.,  we  have 


m'  —  m 

(m'—m)R 

R 

V 

V2 

18.0 

809".65 

44".981 

—  0".065 

0.0042 

18.0 

809  .96 

44  .998 

-0  .048 

0.0023 

18.0 

810  .01 

45  .001 

-0  .045 

0.0020 

18.0 

809  .81 

44  .989 

—  0  .057 

0.00:!2 

18.0 

811  .37 

45  .076 

+  0  .030 

0.0009 

18.0 

811  .28 

45  .071 

+  0  .025 

0.000(5 

18.0 

811  .73 

45  .096 

+  0  .050 

0.0025 

18.0 

811  .85 

45  .103 

+  0  .057 

0.00)52 

18.0 

811  .61 

45  .089 

+  0  .043 

0.0018 

18.0 

811  .04 

45  .058 

+  0  .012 

0.0001 

.8  =  45  .046 
Probable  error  *  =  ±  0.67 


=  0.0208 


.0208 


10X9 


±  0".010. 


*See  Appendix  C,  §  1. 


THE   MICROMETER  77 

By  reducing  the  whole  series  of  transits  the  value  of  R 
and  its  probable  error  were  found  to  be 

R  =  45".059  ±  0".006. 

From  the  level  readings  b  =  +  5".17,  bf  =  +  7".05;  and 
from  observations  for  azimuth  on  /3  Cassiopeice  and  4  H. 
Draconis,  and  on  6  Bootis  and  36  H.  Cassiopeice,  a=  —9".  15, 
a'  =  —  8". 79.  Substituting  these  in  (161)  and  dividing 
by  46,  the  difference  of  the  first  and  last  micrometer  read- 
ings of  the  original  series,  we  have  as  a  correction  to  jR, 
-  0".017,  and  therefore 

R  =  45". 042  ±  0".006. 

There  is  an  indication  from  the  individual  results  for  R 
that  its  value  increases  as  the  micrometer  readings  in- 
crease. This  irregularity  should  be  fully  investigated  by 
further  observations,  and  allowed  for  in  refined  observa- 
tions if  it  proves  to  be  real. 

The  value  of  a  revolution  is  affected  by  changes  of  tem- 
perature. To  determine  the  rate  of  change,  observations 
should  be  made  on  several  nights  at  widely  different  tem- 
peratures. If  R  is  the  value  of  a  revolution  at  the  temper- 
ature r,  RQ  the  value  at  the  temperature  50°,  and  x  the 
correction  to  RQ  for  a  rise  of  1°  in  temperature,  each  night's 
observations  furnish  an  equation  of  the  form 

R  =  R0  +  (r-W)x.  (162) 

The  solution  of  these  equations  by  the  method  of  least 
squares  gives  the  most  probable  values  of  RQ  and  #,  and 
therefore  of  R. 

(c)  If  the  micrometer  is  designed  for  the  measurement 
of  zenith  distances,  the  micrometer  wire  being  horizontal, 
the  observations  are  made  at  the  time  of  the  star's  greatest 
western  or  eastern  elongation.  This  occurs  when  the  ver- 
tical circle  of  the  star  is  tangent  to  its  diurnal  circle.  At 
this  time  the  micrometer  wire  is  parallel  to  the  star's  hour 
circle.  If  w0,  tQ  and  TQ  refer  to  the  instant  of  greatest 


78  PRACTICAL   ASTRONOMY 

elongation  and  w,  t  and  T  to  any  other  instant,  the 
formula  (159)  is  applicable  to  this  case.  At  the  instant 
of  greatest  elongation  the  parallactic  angle  ZOP,  Fig.  1,  is 
90°  for  western  and  270°  for  eastern  elongation,  and  we 
can  write 

cos  t0  =  tan  (f>  cot  8,     cos  z0  =  sin  <f>  cosec  8,     T0  =  a  +  t0.     (163) 

Set  the  telescope  at  the  zenith  distance  20  when  the  star 
is  just  entering  the  instrument.  Note  the  time  of  transit 
over  the  micrometer  wire  ;  and,  as  before,  carry  the  ob- 
servations nearly  across  the  field.  Any  change  in  the 
zenith  distance,  of  the  telescope  a  during  the  progress  of 
the  observations  will  affect  the  resulting  value  of  R.  The 
amount  of  the  change  will  be  indicated  by  the  zenith  level 
and  can  be  allowed  for.  The  level  should  be  read  after 
each  transit  is  observed/  If  1Q  is  the  level  reading,  i.e.  the 
reading  of  the  level  scale  for  the  middle  of  the  bubble,  at 
the  time  TQ,  I  the  level  reading  at  the  time  T,  and  d  the 
value  in  arc  of  a  division  of  the  level,  we  have 


(m  -  m^R  =  ±  sin(r-  ^ 

and  for  another  observation 

(m'  -  m0)R=±sm(T'  -  T 
Whence 

(m'  -m)R  =  ±sin  (T  -  T0)-^l^  sin(3P-  T0)--    +  (I'-Qrf,    (165) 
sin  J.  sin  J. 

in  which  the  lower  sign  is  for  eastern  elongation.  The 
micrometer  readings  are  supposed  to  increase  with  the 
time  for  western  elongations,  and  the  level  readings  to 
increase  towards  the  north. 

The  resulting  value  of  R  must  be  corrected  for  refrac- 
tion. From  the  values  of  ZQ  and  R,  the  zenith  distances 
corresponding  to  the  first  and  last  observations  can  be 
obtained,  and  thence  the  refractions.  The  difference  of 
the  refractions  divided  by  the  difference  of  the  first  and 


THE   LEVEL  79 

last  micrometer  readings  is  the  amount  by  which  the  value 
of  R  must  be  decreased. 

If  both  R  and  d  are  unknown  a  close  approximation  to 
the  value  of  R  is  obtained  by  neglecting  the  term  (lr  —  l)d. 
With  this  value  of  R  the  value  of  d  is  computed  (§  63) 
and  substituted  in  (165),  and  the  corrected  value  of  R 
obtained.  A  second  approximation  to  the  value  of  d  will 
rarely  be  required. 

THE  LEVEL 

62.  The  spirit  level  consists  of  a  sealed  glass  tube, 
ground  on  the  upper  interior  surface  to  the  arc  of  a  circle 
of  large  radius,  and  nearly  filled  with  alcohol  or  ether. 
The  bubble  of  air  occupying  the  space  not  filled  by  the 
liquid  is  always  at  the  highest  point  of  the  curve.  There- 
fore a  change  in  the  relative  elevations  of  the  ends  of  the 
tube  causes  a  motion  of  the  bubble,  the  amount  of  which  is 
read  from  a  scale  marked  on  the  surface  of  the  glass.  The 
level  is  adapted  to  the  determination  of  the  angle  which  a 
nearly  horizontal  line  makes,  with  the  horizon,  or  the  very 
small  angle  moved  over  by  a  telescope. 

The  level  tube  is  mounted  and  attached  to  astronomical 
instruments  in  various  ways,  but  there  is  one  general 
method  of  using  it.  Let  the  divisions  of  the  scale  be  num- 
bered in  both  directions  from  zero  at  the  center,  and  let  d 
be  the  angular  value  of  one  division.  If  the  level  be  placed 
on  a  truly  horizontal  line  —  say,  for  convenience,  an  east 
and  west  line  —  the  center  of  the  bubble  will  not  be  at  zero, 
owing  to  the  non-adjustment  of  the  level.  If  the  cen- 
ter is  x  divisions  from  the  zero,  the  error  of  the  level  is 
dx.  Let  the  level  be  placed  on  a  line  inclined  to  the  hori- 
zon at  an  angle  6,  and  let  the  reading  of  the  west  end  of 
the  bubble  be  w  and  the  east  end  e.  Then  the  elevation  of 
the  west  end  of  the  line  is  given  by 

6  =  \(w  -  e)d  =p  dx. 


80  PRACTICAL   ASTRONOMY 

Now  let  the  level  be  reversed  in  direction  and  let  the  read 
ing  of  the  west  end  be  ivr  and  the  east  end  ef.  Then 

b  =  J(«;'  -  e')d  ±  dx. 

Combining  these  values  of  b  we  have 

&  =  iCO  +  O  -  0  +  e')-\d ;  (166) 

from  which  it  appears  that  the  error  of  the  level  is  elimi- 
nated by  reversing.  A  positive  value  of  b  will  indicate 
that  the  west  end  of  the  line  is  higher  than  the  east  end. 

Whenever  it  is  possible  the  level  should  be  read  several 
times,  the  same  number  of  readings  being  made  in  each 
position,  —  level  direct  and  level  reversed,  —  care  being 
taken  to  remove  the  level  from  its  bearings  after  each 
reading  is  made. 

Example.  The  inclination  of  the  axis  of  a  transit  instru- 
ment is  required  from  the  following  level  readings,  the 
value  of  one  division  being  l;/,88. 


w 

e 

Direct        14.1 

9.7 

Sw       53.4 

Reversed    12.6 

11.1 

2e       41.8 

Reversed    12.7 

11.1 

*  8)  +11.6 

Direct        14.0 

9.9 

+    1.45 

Sum  53.4        41.8 

The  axis  makes  an  angle  -f  1.45  d  =  +  2".73  with  the 
horizon,  the  west  end  being  higher  than  the  east. 

In  case  the  zero  of  the  scale  is  at  one  end  of  the  tube 
and  the  numbers  increase  continuously  to  the  other,  — 
which  is  a  better  system,  —  we  can  show  that 

b  =  i[(w>  +  e}-  (w'  +  e')]rf,  (167) 


in  which  the  readings  w  and  e  for  level  direct  correspond 
to  that  position  of  the  level  for  which  the  readings  increase 
toward  the  west. 

*  It  must  be  noticed  that  there  are  two  complete  observations  for  de- 
termining &,  and  hence  the  divisor  is  8  instead  of  4. 


THE  LEVEL 


81 


Reversed 

w'          16.3 

95.3 

e'          39.4 

111,6 

w'          16.4 

8)  -16.3 

e1          39.5 

-  2.037 

Sum        111.6 

Example.  Find  the  inclination  of  the  axis  of  a  transit 
instrument  from  the  following  level  readings,  the  value  of 
one  division  being  2". 743. 

Direct 
w  35.4 
e  12.4 
w  35.3 
e  12.2 
Sum  95.3 

The  axis  makes  an  angle  —  2.037  d  =  —  5".59  with  the 
horizon,  the  west  end  being  lower  than  the  east. 

63.  The  value  of  one  division  of  the  level  is  determined 
best  by  means  of  a  level-trier.  This  consists  of  a  hori- 
zontal bar  supported  at  one  end  by  two  bearings  and  at 
the  other  by  a  vertical  micrometer  screw.  The  level  is 
placed  on  the  bar  and  the  readings  of  the  micrometer 
and  bubble  are  noted.  The  screw  is  now  turned  and  the 
bubble  moves  to  a  new  position.  The  readings  of  the 
micrometer  and  bubble  are  again  noted.  The  angle  moved 
over  by  the  bar  is  known  from  the  length  of  the  bar,  the 
pitch  of  the  screw  and  the  difference  of  the  micrometer 
readings;  whence  the  angular  value  of  one  division  of  the 
level  may  be  obtained.  If  possible,  the  determination  of 
the  value  of  a  division  should  be  made  after  the  level  tube 
is  fixed  in  its  final  mounting,  rather  than  before. 

The  essential  principles  of  the  level-trier  are  well  illus- 
trated by  Fig.  14. 


FIG.  14 

In  the  absence  of  a  level-trier,  an  accurate  determination 
of  the  value  of  a  division  can  be  obtained  by  means  of  any 
telescope  provided  with  a  micrometer  in  zenith  distance. 
To  illustrate,  let  an  equatorial  be  directed  upon  a  distant 


82 


PRACTICAL   ASTRONOMY 


terrestrial  mark  directly  north  or  south  of  it,  and  adjust  the 
micrometer  wire  to  parallelism  with  the  horizon.  Mount 
the  level  upon  the  telescope  so  that  the  vertical  plane  pass- 
ing through  the  axis  of  the  level  tube  is  parallel  to  the 
line  of  sight,  and  so  that  the  bubble  is  at  one  end  of  the 
scale.  The  mark  is  bisected  by  the  micrometer  wire,  and 
the  level  and  micrometer  readings  noted.  The  instrument 
is  then  turned  through  an  angle  such  that  the  bubble 
moves  to  the  other  end  of  the  scale.  The  mark  is  again 
bisected  by  the  wire,  and  the  level  and  micrometer  read- 
ings noted  as  before.  The  difference  of  the  level  readings 
corresponds  to  the  difference  of  the  micrometer  readings, 
whence  the  value  of  one  division  of  the  level  can  be 
obtained  from  the  known  value  of  a  revolution  of  the 
micrometer  screw.  In  general,  such  observations  are  best 
made  on  an  overcast  day. 

Example.  The  following  observations  were  made  Feb- 
ruary 19,  1891,  to  determine  the  value  of  a  division  of  the 
striding  level  of  the  Detroit  Observatory  transit  instru- 
ment, the  telescope  being  directed  to  a  distant  mark.  The 
value  of  one  revolution  of  the  screw  is  45". 042.  Find  the 
value  of  a  division  of  the  level. 


LEVEL 

DIFFERENCES 

d 

n 

s 

Level 

Micrometer 

20.9 
2.0 

1.1 
20.0 

17.019 
17.791 

18.9 

0.772 

0.0408  R 

1.8 
20.6 

20.2 
1.4 

17.773 

1(1969 

18.8 

0,804 

0.0428  R 

The  mean  of  eighteen  observations  gave  d—  0.041772 
±  0.0004  R  =  1".878  ±  0".018. 

The  level  tube  should  be  thoroughly  tested  for  irregu- 
larity of  curvature  before  using.  If  different  portions  of  a 


THE   CHRONOMETER  83 

level  give  sensibly  different  values  for  a  division  of  the 
scale,  it  should  not  be  used  in  refined  observations. 

The  value  of  a  division  should  also  be  determined  at  two 
or  more  very  different  temperatures  in  order  that  a  tem- 
perature correction  may  be  introduced  if  necessary. 

A  level  should  be  adjusted  by  the  vertical  adjusting 
screws  so  that  the  bubble  will  stand  near  the  center  of  the 
tube  when  the  level  is  placed  on  a  horizontal  line.  It 
should  be  adjusted  by  the  horizontal  screws  so  that  the 
axis  of  the  tube  will  be  parallel  to  the  line  whose  inclina- 
tion is  to  be  measured.  This  adjustment  is  tested  by 
revolving  the  level  slightly  about  its  bearings.  If  the 
readings  are  different  when  the  level  is  equally  displaced 
in  opposite  directions  from  the  vertical  plane  through  its 
bearings,  the  adjustment  is  not  perfect. 

THE   CHRONOMETER 

64.  A  chronometer  is  a  large  and  carefully  constructed 
watch  which  is  "  compensated "  so  that  changes  of  tem- 
perature have  very  little  effect  on  the  time  in  which  the 
balance-wheel  vibrates.  It  is  a  very  accurate  time-piece 
when  properly  handled,  comparing  favorably  with  the 
astronomical  clock,  and  being  portable  is  adapted  to  field 
work  and  navigation. 

The  chronometer  correction  is  the  amount  which  must  be 
added  to  the  reading  of  the  chronometer  face  to  obtain  the 
correct  time.  It  is  +  when  the  chronometer  is  slow.  The 
chronometer  rate  is  the  daily  increase  of  the  chronometer 
correction.  It  is  -j-  when  the  chronometer  is  losing.  It  is 
not  necessary  that  the  correction  and  rate  be  small,  though 
it  is  convenient  to  have  the  rate  less  than  ±55  a  day.  The 
test  of  a  good  time-piece  lies  in  the  uniformity  of  its  rate. 
The  correction  is  generally  allowed  to  increase  indefinitely. 

The  chronometer  correction  is  obtained  by  observations 
on  the  celestial  objects,  or  by  comparison  with  a  time-piece 
whose  correction  is  known.  If 


84  PRACTICAL   ASTRONOMY 

&T0  =  the  chronometer  correction  at  a  time  Tw 
AT1  =  the  chronometer  correction  at  a  time  T, 
$T  =  the  chronometer  rate, 

we  determine  the  rate  per  unit  of  time  by 

87'  =  A!r^-  (168) 

2  ~  2o 

Conversely,  if  the  rate  and  the  correction  at  the  instant 
TQ  are  known,  the  correction  at  the  instant  T  is  given  by 


T0).  (169) 

Example.  The  correction  to  chronometer  T.  S.  &  J.  D. 
Negus,  no.  721,  was  -f  16m  19*.5  at  Ann  Arbor  mean  time 
1891  March  25rf  11*,  and  +  16m  555.6  at  1891  April  4d  11*. 
Find  the  daily  rate,  and  the  correction  at  1891  March 
28d  13*. 

Frbm  (168)  we  find  the  daily  rate  ST  =  +  3S.61.  Sub- 
stituting this  and  T=  March  2Sd  13*  in  (169)  we  find 

AT=  +  16™  198.5  +  38.61  x  3.08  =  +  16W  30«.6. 

The  above  equations  are  true  only  when  the  rate  is  con- 
stant for  the  interval  T—  T0.  Such  constancy  can  be 
assumed  for  an  interval  of  a  few  days  in  the  case  of  the 
best  chronometers  ;  but  when  great  accuracy  is  required 
the  interval  between  observations  for  determining  chro- 
nometer correction  should  be  as  small  as  possible. 

When  several  chronometers  are  employed,  the  correction 
to  one  is  obtained  by  observation  ;  and  to  the  others,  by 
comparison  with  the  first.  If  two  chronometers  which 
keep  the  same  kind  of  time  are  compared,  it  will  generally 
happen  that  they  do  not  beat  together.  The  fraction  of  a 
second  by  which  one  beats  later  than  the  other  can  be  esti- 
mated after  some  practice  to  within  05.1  or  Os.2,  so  that  the 
correction  can  be  obtained  to  that  degree  of  accuracy  by 
this  method. 

When  a  sidereal  chronometer  is  compared  with  a  mean 


THE   CHRONOMETER  85 

time  chronometer  the  degree  of  accuracy  is  higher.  If  the 
chronometers  tick  half  seconds,  the  beats  of  the  two  will 
coincide  once  in  every  183s,  since  in  this  interval  sidereal 
time  gains  05.5  on  mean  solar.  The  ear  is  capable  of  esti- 
mating the  coincidence  of  the  beats  within  05.02  or  0s. 03. 
When  the  coincidence  occurs  the  observer  notes  the  times 
indicated  by  the  two  chronometers.  The  correction  to  the 
one  being  known,  a  satisfactory  value  of  the  correction  to 
the  other  is  readily  obtained. 

When  a  chronograph  [§  68]  is  at  hand  and  the  chronom- 
eters are  provided  with  break-circuits  (or  make-circuits), 
the  comparisons  are  most  conveniently  and  accurately 
made  by  placing  the  two  chronometers  in  the  chronograph 
circuit.  The  beats  are  recorded  on  the  chronograph  sheet 
and  the  distance  between  them  can  be  measured  very 
accurately  by  means  of  a  scale. 

It  is  convenient  to  use  a  sidereal  chronometer  when 
making  observations  on  the  stars,  planets,  comets,  etc.,  and 
a  mean  time  chronometer  when  making  observations  on 
the  sun. 

65.  The  observer  should  be  able  to  "carry  the  beat"  of 
the  chronometer ;  that  is,  mentally  to  count  the  successive 
seconds  from  the  tick  of  the  chronometer  without  look- 
ing at  it.     An  experienced  observer  will  carry  the  beat 
for  several  minutes,  estimate  the  times  of   transits  of  a 
star  over  several  wires  (or  other  similar  phenomena)  to 
tenths  of  seconds,  and  write  them  on  a  slip  of  paper  with- 
out taking  his  eye  from  the  telescope :  then,  still  carrying 
the  beat,  he  will  look  at  the  chronometer  face  to  verify  his 
count.     This  is  called  the  "  eye  and  ear  method  "  of  observ- 
ing, and  it  is  very  important  that  every  observer  should  be 
able  to  employ  it  with  accuracy  and  perfect  ease. 

66.  To  obtain  the  best  results  from  a  chronometer  the 
following  precepts  should  be  rigidly  observed  : 

(a)  It  should  be  wound  at  regular  intervals.     If  it  re- 


86  PRACTICAL   ASTRONOMY 

quires  winding  daily  it  should  always  be  wound  at  the 
same  hour  of  the  day;  otherwise  an  unused  part  of  the 
spring  is  brought  into  action  and  a  change  of  rate  results. 
(£)  The  hands  should  not  be  moved  forward  oftener 
than  is  necessary,  and  they  should  not  be  moved  backward. 

(c)  A  chronometer  on  shipboard  should  be  allowed  to 
swing  freely  in  its  gimbals,  so  that  it  may  always  take  a 
horizontal  position;    but  when  carried  about  on  land  it 
should  be  clamped  so  as  to  avoid  the  violent  oscillations 
due  to  the  sudden  motions  it  receives. 

(d)  It  should  be  kept  in  a  dry  place;  as  nearly  at  a 
uniform  temperature  as  possible ;  away  from  magnetic  in- 
fluences ;  and  when  at  rest  should  always  be  in  the  same 
position  with  respect  to  the  points  of  the  compass. 

(e)  All  quick  motions  should  be  avoided:  in  particular, 
it  should  never  be  rotated  rapidly  about  its  vertical  axis. 

(jQ  In  out-of-door  use  it  should  be  protected  from  the 
direct  rays  of  the  sun. 

67.  The  astronomical  clock  is  a  finely  constructed  clock 
whose  pendulum  is  compensated  for  changes  of  tempera- 
ture.    Its  rate  is  in  general  more  uniform  than  that  of  a 
chronometer.     It  is  one  of  the  fixed  instruments  of  an 
observatory,  and  to  that  extent  the  remarks  concerning 
the  chronometer  are  applicable  to  it. 

THE   CHRONOGRAPH 

68.  The  chronograph  is  a  mechanical  device  for  recording 
the  instant  when  an  observation  is  made.   A  sheet  of  paper 
on  which  the  record  is  to  be  made  is  wrapped  around  a 
metallic  cylinder  which  is  caused  to  rotate  once  per  minute 
by  means  of  clock-work.    A  pen  is  attached  to  the  armature 
of  an  electro-magnet  in  such  a  way  as  to  press  its  point  on 
the  moving  paper.    The  magnet  is  carried  slowly  along  the 
cylinder  by  a  screw,  so  that  the  pen  traces  a  continuous 
spiral  on  the  paper.     The  electro-magnet  is  placed  in  an 


88  PRACTICAL  ASTRONOMY 

electric  circuit  which  passes  through  the  chronometer  or 
clock  (or,  better,  through  a  relay  connected  with  the  time- 
piece), iu  such  a  way  that  the  circuit  is  broken  for  an 
instant  at  the  beginning  of  every  second,  or  every  other 
second.  At  each  of  these  instants  the  electro-magnet 
releases  the  armature  carrying  the  pen,  the  pen  moves 
laterally  for  the  moment,  and  in  this  way  the  spiral  is 
graduated  by  notches  to  seconds  of  time.  One  notch  is 
usually  omitted  at  the  beginning  of  each  minute,  to  assist 
in  identifying  the  seconds.  One  of  the  circuit  wires  passes 
through  a  signal-key  held  in  the  observer's  hand.  When  a 
star,  for  example,  is  being  observed  he  presses  the  key  at 
the  exact  instant  when  the  star  is  crossing  a  wire,  thus 
breaking  the  circuit  and  making  the  record  on  the  chrono- 
graph sheet.  The  beats  of  the  chronometer  being  recorded 
on  the  sheet,  the  chronometer  time  when  the  key  was 
pressed  can  be  read  from  the  sheet  by  means  of  a  scale 
with  great  accuracy,  and  at  the  observer's  leisure. 

When  the  chronograph  is  first  set  in  motion  the  observer 
records  in  his  note-book  the  hour,  minute,  and  second  cor- 
responding to  a  certain  marked  notch  on  the  sheet,  which 
serves  as  a  reference  point  in  identifying  all  the  notches 
on  the  sheet. 

In  some  forms  of  the  chronograph  the  circuit  is  made  by 
pressing  the  key,  but  the  break-circuit  is  preferable. 

The  chronographic  method  is  generally  preferable  to  the 
eye  and  ear  method  because  it  relieves  the  mind  from 
carrying  the  beat  and  making  the  record,  thus  allowing 
greater  care  to  be  given  to  other  parts  of  the  observation, 
and  because  more  observations  can  be  made  in  a  given  time. 
Rut  in  the  case  of  transit  observations  of  slowly-moving 
stars,  or  of  very  faint  objects,  and  in  many  forms  of  microm- 
eter observations,  the  eye  and  ear  method  is  at  least  as 
satisfactory  as  the  chronographic  method. 

A  very  common  form  of  chronograph  is  illustrated  in 
Fig.  15. 


CHAPTER  VI 


THE   SEXTANT 

69.   The  sextant  is  an  instrument  especially  adapted  to 
the  determination  of  time,  latitude  and   longitude  when 
extreme  accuracy  is  not  required,  as  in  navigation  and 
exploration.     It  con- 
sists essentially  of  a 
brass     frame     ADO, 
Fig.    16,    bearing    a 
graduated  arc  AC,  a 
telescope  EF,  whose 
line  of  sight  is  paral- 
lel  to    the    plane    of 
the     graduated     arc, 
and    the    mirrors  H 
and  D,  whose  planes 
are  perpendicular  to 
the  plane  of  the  arc. 
The  mirror  D,  called 
the     index-glass,     is 
iixed    to    the    index- 
arm  DB,  which  revolves  about  D  at  the  center  of  the  arc, 
and  which  carries  a  vernier  at  B.     The  mirror  H,  called 
the  horizon-glass,  is  attached  to  the  frame.     The  lower 
half  of  it  is  silvered,  the  upper  half  is  left  clear. 

Figure  17  illustrates  a  form  of  sextant  commonly  em- 
ployed. The  special  parts  already  described  in  connec- 
tion with  Fig.  16  will  be  recognized  without  difficulty. 
The  telescope  is  mounted  on  an  adjustable  standard  so 

89 


FIG.  16 


90 


PRACTICAL   ASTRONOMY 


that  its  distance  from  the  frame  of  the  sextant  may  be  varied 
by  turning  the  screw-head  at  the  lower  end  of  the  standard. 
Colored  or  neutral-tint  glasses  are  mounted  in  front  of 
the  index  and  horizon  glasses.  They  can  be  rotated  into  the 
paths  of  the  sun's  rays  to  protect  the  eyes  while  observing 
that  body.  A  dense  neutral-tint  glass  may  also  be  screwed 


FIG.  17 

over  the  eyepiece  for  the  same  purpose.  The  telescope 
may  be  replaced  by  others  of  different  magnifying  power, 
or  by  one  with  a  larger  object-glass  for  observing  stars,  — 
shown  in  the  foreground  of  the  cut.  The  index-arm  car- 
rying the  vernier  is  furnished  with  a  clamp  and  slow 
motion  for  setting  accurately  to  any  desired  reading. 

70.  To  illustrate  the  method  of  using  a  sextant  and  the 
principles  involved,  let  it  be  required  to  measure  the  angle 
SlSSr  between  the  stars  >S  and  S'.  The  instrument  is  held 
in  the  hand  and  the  telescope  directed  to  the  star  &.  The 
ray  SE  passes  through  the  unsilvered  part  of  H  and  forms 


THE   SEXTANT  91 

a  direct  image  of  the  star  at  the  focus  F.  The  sextant  is 
revolved  about  the  line  of  sight  until  its  plane  passes 
through  the  other  star  8'.  The  index-arm  is  then  moved 
until  the  reflected  image  of  8'  is  brought  into  the  field  and 
nearly  in  coincidence  with  the  direct  image  of  8.  The 
index-arm  is  clamped  and  the  two  images  brought  into 
perfect  coincidence  by  turning  the  slow-motion  or  tangent 
screw.  If  the  instrument  is  perfectly  constructed  and 
adjusted,  the  required  angle  is  given  at  once  by  the  circle 
reading.  The  ray  of  light  8' D,  which  forms  the  reflected 
image  at  F,  traverses  the  path  8' D-DH-HF,  being  re- 
flected by  the  two  mirrors  D  and  H.  When  the  direct 
and  reflected  images  coincide,  the  angle  between  the 
stars  is  twice  the  angle  between  the  mirrors.  That  is, 
8E8'  =  Z  ELD,  since  the  angle 

SES'  =  180°  -  EDH  -  EHD 

=  180°  -  2  HDL  -  2  (LHD  -  90°) 
=  2  (180°  -  HDL  -  LHD) 
=  2  ELD. 

If  A  is  the  position  of  the  zero  of  the  vernier  when  the 
two  mirrors  are  parallel,  and  B  its  position  when  the  two 
images  coincide,  we  have 

SES'  =  2HLD  =  2ADB  =  2AB.  (170) 

It  thus  appears  that  to  enable  us  to  read  the  required  angle 
directly  from  the  circle,  the  circle  reading  must  be  twice 
the  corresponding  arc.  Thus,  the  120°  line  is  really  only 
60°  from  the  0°  line  [or  a  sextant,  hence  the  name]. 

An  improved  form  of  the  sextant  is  known  as  the  Pistor 
and  Martins  (Berlin)  prismatic  sextant,  in  which  the  hori- 
zon glass  is  replaced  by  a  totally  reflecting  prism,  occupying 
a  somewhat  different  position  on  the  frame  of  the  instru- 
ment. Among  other  advantages  of  the  prismatic  form  it 
can  be  used  for  measuring  angles  up  to  180°  and  even 


92  PRACTICAL   ASTRONOMY 

greater ;  whereas  with  the  common  form,  the  angle  is  lim- 
ited to  about  140°. 

Again,  the  graduated  arc  is  sometimes  a  complete  circle, 
in  which  case  the  index  arm  is  extended  over  a  diameter 
of  the  circle  and  carries  a  vernier  on  each  extremity.  Such 
an  instrument  is  called  a  reflecting  circle  or  prismatic  circle 
according  as  a  horizon-glass  or  prism  is  used.  Its  chief 
advantage  lies  in  the  fact  that  the  eccentricity  is  eliminated 
by  the  use  of  two  verniers  180°  apart. 

71.  In  order  to  obtain  good  results  with  the  sextant, 
the  instrument  must  be  accurately  adjusted,  and  the  tele- 
scope focused;  the  direct  and  reflected  images  should  be 
about  equally  bright;  and  several  complete  observations 
should  be  made,  the  mean  of  all  being  used. 

The  images  are  made  equally  bright  by  moving  the  tele- 
scope from  or  toward  the  frame,  so  as  to  utilize  more  or 
less  of  the  light  passing  through  the  transparent  part  of 
the  horizon-glass,  or  by  placing  colored-glass  shades  in 
front  of  the  index-glass. 

In  measuring  the  angular  distance  between  two  stars, 
the  images  of  the  stars  are  brought  into  exact  coincidence 
in  the  middle  of  the  field  of  view.  In  measuring  the  dis- 
tance of  the  moon  from  a  star,  the  star  is  brought  into 
coincidence  with  that  point  of  the  moon's  bright  limb 
which  lies  in  the  great  circle  joining  the  star  and  the 
center  of  the  moon.  The  measured  distance  is  then  in- 
creased or  decreased  by  the  moon's  semidiameter  [§§  33, 
34,  35],  In  the  case  of  the  sun  and  moon  the  images  of 
the  nearest  limbs  are  made  to  coincide,  and  the  measured 
distance  is  increased  by  the  semidiameters  of  both  objects, 
as  before.  Results  obtained  in  this  way,  when  corrected 
for  any  instrumental  errors,  are  the  apparent  distances  be- 
tween the  objects. 

The  sextant  is  also  used  for  measuring  the  apparent  alti- 
tudes of  the  heavenly  bodies.  At  sea  the  telescope  is 


THE   SEXTANT  93 

directed  to  that  point  of  the  horizon  which  is  below  the 
object.  The  reflected  image  is  brought  into  contact  with 
the  horizon  line.  When  the  instrument  is  vibrated  slightly 
about  the  line  of  sight  the  image  should  describe  a  curve 
tangent  to  the  horizon.  The  sextant  reading  corrected  for 
instrumental  errors  and  the  dip  of  the  horizon  [§  32]  is  the 
apparent  altitude.  If  the  object  is  the  sun,  the  lower  or 
upper  limb  is  made  tangent  to  the  horizon ;  if  the  moon, 
the  bright  limb ;  and  the  sextant  readings  must  be  further 
corrected  for  semidiameter. 

For  observing  altitudes  on  land  an  artificial  horizon  is 
used.  This  is  a  shallow  basin  of  mercury  over  which  is 
placed  a  roof,  made  of  two  plates  of  glass  set  at  right 
angles  to  each  other  in  a  frame,  to  protect  the  mercury 
from  agitation  by  air  currents.  The  mercury  forms  a  very 
perfect  horizontal  mirror  which  reflects  the  rays  of  light 
from  the  star.  If  the  observer  places  his  eye  at  some  point 
in  a  reflected  ray,  he  will  see  an  image  of  the  star  in  the 
mercury,  whose  angle  of  depression  below  the  horizon  is 
equal  to  the  altitude  of  the  star  above  the  horizon.  If 
then  he  directs  the  telescope  to  the  image  in  the  mercury, 
and  brings  the  two  images  into  coincidence  as  before,  the 
sextant  reading  corrected  for  instrumental  errors  is  double 
the  apparent  altitude  of  the  star.  The  sun's  altitude  is 
measured  by  making  the  two  images  tangent  externally 
The  corrected  sextant  reading  is  double  the  altitude  of 
the  lower  or  upper  limb,  according  as  the  nearest  or  far- 
thest limbs  of  the  sun  and  its  image  in  the  mercury  are 
observed. 

The  double  altitudes  of  stars  near  the  meridian  are 
changing  slowly,  and  the  images  are  brought  into  contact 
by  means  of  the  slow-motion  screw  as  before.  But  the 
double  altitudes  of  stars  at  a  distance  from  the  meridian 
are  changing  rapidly,  and  another  method  is  used.  To 
illustrate,  suppose  the  sun  is  observed  for  time  when  it  is 
east  of  the  meridian,  and  the  altitude  therefore  increasing 


94  PRACTICAL   ASTRONOMY 

The  upper  limb  is  observed  first.  The  two  images  are 
brought  into  the  field  and  the  index  moved  forward  until 
the  sextant  reading  is  from  10'  to  20'  greater  than  the 
double  altitude  of  the  upper  limb,  and  the  instrument  is 
clamped.  The  images  are  now  slightly  separated,  but  they 
are  approaching.  When  they  become  tangent,  the  ob- 
server notes  the  time  on  the  chronometer  and  reads  the 
circle.  The  index  is  again  moved  forward  from  10'  to  20' 
and  the  contact  observed  as  beforec  In  this  way,  four  or 
five  observations  are  made.  The  double  diameter  of  the 
sun  is  about  64',  and  for  observing  the  lower  limb  the 
index  is  quickly  moved  backward  about  45'.  The  two 
images  now  overlap,  but  they  are  separating,  and  the  time 
is  noted  when  they  become  tangent.  Moving  the  index 
forward  as  before,  four  or  five  observations  are  made  on 
the  lower  limb. 

If  the  sun  is  observed  west  of  the  meridian,  the  altitudes 
of  the  lower  limb  should  be  measured  first. 

72.  The  faces  of  the  glass  in  the  horizon  roof  should  be 
perfectly  parallel.     If  they  are  prismatic  the  observed  alti- 
tudes are  erroneous.     The  error  is  eliminated  by  observing 
one-half  of  a  set  of  altitudes  with  the  roof  in  one  position 
and  the  other  half  with  the  roof  in  the  reversed  position, 
and  taking  the  mean  of  all.     Likewise,  the  glass  screens 
in   front   of   the   index   and   horizon   glasses   must   have 
parallel  faces. 

The  surface  of  the  mercury  can  be  freed  from  impurities 
by  adding  a  little  tin-foil.  The  amalgam  which  forms  can 
be  drawn  to  one  side  of  the  basin  by  means  of  a  card,  leav- 
ing a  perfectly  bright  surface. 

ADJUSTMENTS    OF   THE   SEXTANT 

73.  (#)    The  index-glass.     Place  the  sextant  on  a  table, 
unscrew  the  telescope  and  set  it  in  a  vertical  position  on 
the  graduated  arc.     Place  the  eye  near  the  index-glass 


ADJUSTMENTS    OF    THE    SEXTANT  95 

and  move  the  index-arm  toward  the  telescope  until  the 
telescope  and  its  image  in  the  mirror  are  seen  very  nearly 
in  coincidence.  Their  corresponding  outlines  will  be  par- 
allel if  the  index-glass  is  perpendicular  to  the  plane  of  the 
arc.  If  they  are  not  parallel,  the  glass  is  removed  and  one 
of  the  points  against  which  it  rests  is  filed  down  the  proper 
amount.  The  axis  of  the  telescope  is  here  assumed  to  be 
perpendicular  to  the  plane  of  the  end  on  which  it  rests. 
This  can  be  tested  by  rotating  the  telescope  about  its  axis 
and  noticing  whether  the  angle  between  the  tube  and  its 
image  varies.  The  telescope  should  be  set  at  the  mean  of 
the  two  positions  which  give  the  maximum  and  minimum 
values  of  this  angle.* 

(£)  The  horizon-glass.  The  index-glass  having  been 
adjusted,  the  telescope  is  directed  to  a  star  and  the  index- 
arm  is  brought  near  the  zero  of  the  arc.  If  the  horizon- 
glass  is  parallel  to  the  index-glass  the  reflected  image  will 
pass  through  the  direct  image  when  the  index-arm  is  moved 
slowly  to  and  fro.  If  it  passes  on  either  side  of  the  direct 
image  the  horizon-glass  needs  adjustment.  This  is  done 
by  turning  the  screws  provided  for  the  purpose. 

(<?)  The  telescope.  Two  parallel  wires  are  placed  in  the 
telescope  tube.  These  are  made  parallel  to  the  plane  of 
the  sextant  by  revolving  the  tube'  containing  them.  The 
line  of  sight  is  the  line  joining  a  point  midway  between 
these  wires  and  the  center  of  the  object  glass.  This  should 
be  parallel  to  the  plane  of  the  sextant.  To  test  the  adjust- 
ment, select  two  well-defined  objects  about  120°  apart, 
and  bring  the  two  images  into  coincidence  on  one  of  the 
side  wires,  and  then  move  the  sextant  so  as  to  bring  the 
images  on  the  other  wire.  If  the. images  still  coincide, 
the  line  of  sight  needs  no  adjustment.  If  the  images  are 
separated,  the  collar  which  holds  the  telescope  is  shifted 
by  means  of  screws  until  the  adjustment  is  satisfactory. 

*  This  method  was  proposed  by  Professor  J.  M.  Schaeberle :  The 
Sidereal  Messenger,  May,  1888. 


96  PRACTICAL   ASTRONOMY 

CORRECTIONS   TO    SEXTANT    READINGS 

74.  The  index  correction.  It  is  seen  from  (170)  that 
all  angles  measured  with  the  sextant  are  reckoned  from 
A,  the  point  where  the  zero  of  the  vernier  falls  when  the 
two  mirrors  are  parallel ;  whereas  the  circle  readings  are 
measured  from  0°.  The  index  correction  is  the  reading  0° 
(or  360°)  minus  the  reading  at  A.  Let  it  be  represented 
by  /.  The  value  of  /  can  be  reduced  to  zero  by  rotating 
slightly  the  horizon-glass  by  means  of  screws  provided 
for  that  purpose.  But  this  adjustment  is  very  liable  to 
derangement,  and  it  is  customary  to* determine  I  every 
time  the  sextant  is  used  and  apply  it  to  all  the  sextant 
readings. 

(a)  To  determine  /  for  correcting  stellar  observations, 
point  the  telescope  to  a  star  and  bring  the  direct  and 
reflected  images  into  coincidence.  Let  the  sextant  reading 
be  R.  The  index  correction  is  given  by 

/  =  0°  -  #.  (171) 

Example.     Determine  /  from  the  following  readings : 

359°  56'    0"  359°  55'  50" 

56  10  56  10 

55  50  55  55 

The  mean  of  the  six  readings  is  359°  55'  59". 2,  and 

therefore 

/  =  360°  -  359°  55'  59".2  =  +  4'  0".8. 

(5)  For  reducing  solar  observations,  point  the  telescope 
to  the  sun  and  bring  the  direct  and  reflected  images  exter- 
nally tangent  to  each  other  and  read  the  circle.  Then 
move  the  reflected  image  over  the  direct  image  until  they 
are  again  externally  tangent,  and  read  the  circle.  Let  the 
readings  in  the  two  positions  be  Rl  and  Rv  Rl  being  the 
greater.  The  reading  when  the  two  images  coincide  is 
^2)'  an(^  ^ne  iudex  correction  is  given  by 

/  =  360°  -  H^i  +  ^2)-  (172) 


CORRECTIONS    TO    SEXTANT    READINGS  97 

The  observed  semidiameter  of  the  sun  is  given  by 

S  =  $(Rl-R2).  (173) 

To  eliminate  the  effect  of  refraction  the  horizontal  semi- 
diameter  should  be  measured. 

Example.  Find  I  and  S  from  the  following  readings  on 
the  sun  made  Thursday,  1891  April  23. 

360°  28'  35"  359°  24'  50" 
40  40 

45  45 

45  45 

40  50 

Means    360   28  41  .0  359   24  46  .0 

/  =  360°  -  359°  56'  43".5  =  +  3'  16".5. 
S  =  |(l°3'  55".0)  =  15'  58".7. 

From  the  American  Ephemeris,  p.  56,  S  —  15'  56".3. 

75.  Correction  for  eccentricity.  The  arc  of  a  sextant 
being  short,  the  eccentricity  cannot  be  eliminated  by 
means  of  two  verniers  180°  apart,  and  it  must  be  investi- 
gated. This  can  be  done  by  comparing  several  angles 
measured  with  the  sextant  with  their  known  values  ob- 
tained in  some  other  way.  Thus  in  Fig.  11  the  sextant 
reading  is  twice  the  arc  AM.  The  true  value  of  the  angle 
is  obtained  by  correcting  the  reading  at  M  for  eccentricity, 
and  correcting  the  position  of  A  for  eccentricity  and  index 
error.  The  true  reading  at  M  is  given  by  (157).  The 
true  reading  at  the  zero  point  A  is  given  by 

RQ  =  —  I  +  e"  sin  77. 
The  true  value  of  the  angle  is 

R-R0=M  +  c"  sin  (v  +  17)  -  e"  sin  77  +  I.  (174) 

But   R  —  RQ  is  the  known  value  of  the  angle ;  let  d  repre- 
sent it.     M  is  the  observed  value  of  the  angle;  let  d1  rep 


98  PRACTICAL    ASTRONOMY 

resent  it.  Now,  since  an  arc  on  the  sextant  is  one-half  the 
corresponding  reading,  we  have 

i  (d  -  d')  =  e"  sin  (v  +  r?)  -  e"  sin  17  +  £  /, 
which  reduces  to 

d-d'  =  ±e"  cos  Q  v  +  r?)  sin  |  v  +  /  (175) 

=  4  e"  cos  >;  sin  £  v  cos  |  v  —  4  e"  sin  17  sin2  £  v  +  /. 
If  we  put 

4  e"  cos  77  =  xt        4  e"  sin  17  =  y,  (176) 

we  have 

sin  |  v  cos  |  v  a;  —  sin2iry  +  I  —  d  —  d'.  (177) 

This  equation  involves  three  unknown  quantities,  a?,  ?/,  /. 
Three  measured  angles,  each  furnishing  an  equation  of  the 
form  (177),  are  required  for  the  solution  of  the  problem. 

There  are  several  ways  in  which  to  obtain  the  value  of 
d  —  d'  at  any  point  of  the  arc. 

(a)  For  those  who  have  access  to  a  meridian  circle,  the 
most  direct  process  known  is  the  ingenious  method  proposed 
by  Professor  Schaeberle  in  der  Astronomische  Nachrichten, 
no.  2832. 

(6)  When  the  latitude  of  the  observer  and  the  time  are 
accurately  known,  make  a  series  of  measures  of  the  double 
altitudes  of  a  star  just  before  and  after  its  meridian  pas- 
sage. The  observed  double  altitude  at  the  instant  of  tran- 
sit is  obtained  from  these  measures  by  the  method  of  §  87. 
The  apparent  double  altitude  at  the  instant  is  obtained  at 
once  from  the  known  declination,  latitude  and  refraction. 
The  latter  minus  the  former  is  d  —  df. 

(c)  When  the  latitude  and  time  are  not  accurately 
known,  measure  the  distance  between  two  stars  and  com- 
pare it  with  the  known  apparent  distance.  The  apparent 
distance  is  found  by  the  method  of  §  10,  using  a',  Br  and 
a",  &lf  as  affected  by  refraction,  §  31. 

Example.  The  distance  between  Aldebaran  and  Arcturus 
was  measured  with  the  sextant  at  Ann  Arbor,  Thursday 
night,  1891  March  5,  as  below.  It  is  required  to  form  the 


CORRECTIONS   TO    SEXTANT    READINGS 


99 


equation  (177)  for  this  pair  of  stars, 
correction  A0  was  -f  15m  1s. 


The  chronometer 


Chronometer 


Sextant 


8*  37m  25s        130°  14'  55" 


45 

30 

14 

40 

48 

30 

14 

55 

52 

20 

14 

35 

56 

20 

14 

55 

9 

0 

30 

14 

45 

9 

4 

0 

14 

40 

Means 

8 

52 

5 

130  14 

46.4 

A0 

+ 

15 

7 

Barom.     29  .400  inches 
Att.  Therm.     65°.0  F. 
Ext.  Therm.     18°.0  F. 

Amer.  Ephem.,  pp.  322,  340 
A  Idebaran  A  returns 

a     4*  29™  39-  .33     14A  10™  41s  .97 
8  16°  17'  22"  .7       19°  44'  47"  .8 


0  9     7    12 


With  these  data  we  solve  (41),  (35),  (36),  (37),  (32), 
(95),  (100)  and  (101)  as  below. 


A  Idebaran 

A  returns 

sin*    9.97127 

9.98667n 

6    9h    7m  12* 

ga    7™  i2. 

cos<£    9.86915 

9.86915 

a    4   29    39 

14    10  42 

cosecz    0.04642 

0.03729 

t    4   37    33 

18   56  30 

cosecg    0.11316 

0.10689n 

*69°23'  15" 

284°   7'  30" 

log  1    0.00000 

0.00000 

<£42   16    47 

42   16  47 

cot<f>    0.04130 

0.04130 

True  z  63°  58'  41" 

66°  35'  42" 

cos*    9.54660 

9.38746 

Meanrefr.          1    56 

2  11 

£21°    9'  54" 

15°   1'24" 

App.263   56    45 

66°  33  31 

816   17    23 

19  44  48 

log/*    1.75821 

1.75766 

tan*    0.42467 

0.59921n 

tans    0.31078 

0.36292 

sinL    9.55758 

9.41366 

AlogBT   9.99583 

9.99583 

sec(S-fL)    0.10027 

0.08542 

A  logy    0.02746 

0.02750 

tan  £    0.08252 

0.09829n 

logr    2.09228 

2.14391 

q  50°  24'  39" 

308°  34'  16" 

sin  q    9.88684 

9.89311n 

cot(8  +  L)    0.11573 

0.15849 

sec  8    0.01780 

0.02632 

cos  q    9.80433 

9.79482 

log  da.    1.99692 

2.06334n 

z  63°  58'  41" 

66°  35'  42" 

cos  q    9.80433 

9.79482 

d8   +  1'  18".8 

+  1'  26".8 

da   +  1   39  .3 

-  1  55  .7 

Applying  these  refractions  to  the  above  star  places  we 
obtain  the  coordinates  which  are  to  be  used  in  solving 
(53),  (54),  (55)  and  (50). 


100  PKACTICAL    ASTKONOMY 

a!   67°  26'  29".2        cot  (8"  +  G)  9.914333n 

8'   16  18  41  .5  cos£'  9.842600 

a"  212  38  33  .8  d  130°  17' 22".4 
8"   19  46  14  .6 

a"  -  a'  145  12  4  .6        sin  (a"  -  a')  9.756404 

cot  8'    0.533668  cos  8'  9.982158 

cos  (a" -a')    9.914429n  cosec  B'  0.143841 

G  289°36'52".7  cosec  d  0.117597 

sin  G    9.974038n  log!  0.000000 

tan  (a"  -  a')    9.841975* 

sec  (8"  +  G}         0.197546  d'  130°  14'  46".4 

tanB'        0.013559  d-d'  +236.0 

B'      45°  53'  39".3  d-d'  +    156  .0 

The  angle  v  in  (177)  is  not  ^d1,  but  one-half  the  read- 
ing corresponding  to  the  line  of  the  circle  with  which  the 
vernier  line  coincides,  and  it  is  the  eccentricity  of  this  point 
which  enters  into  d—  d'.  For  the  reading  df  =  130°  14'  50" 
the  29th  line  of  the  vernier  coincides  with  the  circle  line 
135°  0',  and  therefore  in  this  case  J  v  =  33°  45'.  We  now 

find 

sin  £  v  cos  \  v  -  0.462,     sin2  \  v  -  0.309  ; 

and  therefore 

0.462  x  -  0.309  y  +  /  =  156.0. 

Similarly,  from  the  meridian   double  altitude   of   a  star, 
method  (6),  and  from  another  pair  of  stars  we  find 

0.259  x  -  0.072  y  -f  7  =  165.0, 
0.117  x-  0.014  y  +  I=  171.0. 

Solving  these  three  equations  we  obtain 

log  a;  =  1.61380B,     logy  =  0.43265,     /  =  175.8; 
whence,  from  (176), 

77  =  176°  14',     4e"  =  41".2. 

While  the  index  correction  varies  from  day  to  day,  and 
its  value  should  be  determined  by  the  methods  of  §  74 
every  time  the  sextant  is  used,  the  eccentricity  is  prac- 
tically constant.  By  neglecting  the  term  /  in  (175)  and 


DETERMINATION    OF    TIME 


101 


making  2  v  successively  0°,  10°,  etc.,  we  obtain  the  follow- 
ing corrections  for  eccentricity  to  be  applied  to  the  circle 
readings. 


Circle 

Correction 

Circle 

Correction 

Circle 

Correction 

0° 

0".0 

50° 

-    8".8 

100° 

-  16".2 

10 

-  1   .8 

60 

-10  .5 

110 

-17  .4 

20 

-3  .6 

70 

-12  .0 

120 

-  18  .5 

30 

-5  .4 

80 

-13  .5 

130 

-  19   .4 

40 

-7  .2 

90 

-14  .9 

140 

-20  .2 

In  order  to  determine  the  eccentricity  very  accurately, 
at  least  ten  known  angles  distributed  uniformly  from  0° 
to  140°  should  be  measured  and  the  resulting  equations 
solved  by  the  method  of  least  squares.  The  observations 
should  be  made  in  one  night,  so  that  /  may  be  considered 
constant;  but  the  observer  should  determine  I  several 
times  during  the  night,  to  make  sure  that  it  does  not 
change. 

DETERMINATION   OF   TIME 

76.  Time  is  determined  from  observations  on  the  heav- 
enly bodies  by  determining  the   corrections  to  the   chro- 
nometer  or   other  time-piece   at  the   instants   when   the 
observations  are  made. 

77.  By  equal  altitudes  of  a  fixed  star.     When  a  star  is 
from  two  to  four  hours  east  of  the  meridian  and  near  the 
prime  vertical,   observe  a  series  of   its   double    altitudes 
[§71]  with  the  sextant  and  sidereal  chronometer,  and  let 
the  mean  of  the  chronometer  times  be  6 '.     When  the  star 
reaches  the  same  altitude  west  of  the  meridian  observe  its 
double  altitudes  with  the  vernier  of  the  sextant  set  at  the 
same  readings  as  before,  in   inverted  order,  and  let  the 
mean  of  the  chronometer  times  be  0".     The  chronometer 
time  of   the  star's 'meridian  passage  is  -J  (61  -\-  6").     The 


102  PRACTICAL   ASTRONOMY 

sidereal  time  of  the  star's  meridian  passage  equals  its  right 
ascension  a.  The  chronometer  correction  A0  at  this  in- 
stant is  given  by 

0").  (178) 


If  a  mean  time  chronometer  is  employed  the  sidereal 
time  a  must  be  converted  into  mean  time.  The  required 
chronometer  correction  is  then  given  by  (178)  as  before. 

Example.  The  following  equal  altitudes  of  Arcturus, 
were  observed  with  a  sextant  and  sidereal  chronometer  at 
Ann  Arbor,  Saturday  night,  1891  April  25.  Required  the 
chronometer  correction. 

Chronometer  Sextant  Chronometer 

Star  East  reading  Star  West 

10*  23"  21*  81°  30'  0"  17*  21"*  47« 

24  14  81   50  0  20    52 

25  9  82  10  0  19  56 

26  4  82  30  0  19  2 

27  0  82  50  0  18  7 
&  10  25  9.6  0"  17   19  56.8 

Amer.  Ephem.,  p.  340,  a    14*  10"  42S.8 

£(0'  +  0")     13    52    33.2 

A0       +18      9  .6 

78.  By  equal  altitudes  of  the  sun.  Observe  as  described 
above  [§§  71,  77]  the  two  series  of  equal  double  altitudes 
of  the  sun  before  and  after  noon,  and  let  the  chronometer 
times  of  the  east  and  west  observations  be  T1  and  T",  a 
mean  time-piece  being  used.  The  mean  of  the  two  times 
is  not  the  chronometer  time  of  the  sun's  meridian  passage, 
since  the  sun's  declination  has  changed  during  the  interval, 
and  a  correction  must  be  applied.  To  find  its  value  let 

t  =  half  the  interval  between  the  observations  =  \(T"  —  T'), 
8  =  the  sun's  declination  at  the  observer's  apparent  noon, 
d8  =  the  increment  of  the  sun's  declination  in  the  interval  t, 
dt  =  the  increment  of  the  sun's  hour  angle  due  to  the  increment  of 
the  declination. 


DETERMINATION    OF   TIME  103 

Differentiating  (15),  regarding  8  and  t  as  variables,  and 
dividing  by  15  to  express  dt  in  seconds  of  time,  we  have 

\</8  (179) 

1  15 


sin  t       tan  1 

by  which  amount  the  east  and  west  observed  times  are 
greater  than  they  would  be  if  the  declination  were  con- 
stant and  equal  to  S.  The  chronometer  time  of  the  sun's 
meridian  passage  is  therefore  \  (T'  +  T")  —  dt.  The  mean 
time  of  the  sun's  meridian  passage  is  HI,  the  equation  of 
time  at  the  observer's  apparent  noon  ;  and  therefore  the 
chronometer  correction  at  the  mean  of  the  two  times  is 

±T  =  E  -%(T'  +  T")  +  dt.  (180) 

If  a  sidereal  chronometer  is  employed  the  sidereal  inter- 
val t  is  converted  into  the  equivalent  mean  interval  [§  16], 
dt  is  computed  from  (179)  as  before  and  subtracted  from 
the  mean  of  the  two  times,  and  the  result  is  the  chronome- 
ter time  of  the  sun's  meridian  passage.  The  sidereal  time 
at  this  instant  is  equal  to  the  sun's  apparent  right  ascen- 
sion a,  and  the  chronometer  correction  is  given  by 

A0  =  a  -  i  (0'  +  6")  +  dt.  (181) 

Example.  The  equal  double  altitudes  of  the  sun  were 
observed  as  below  at  Ann  Arbor,  Saturday,  1891  April  24- 
25,  a  sidereal  chronometer  being  used.  Find  the  chro- 
nometer correction. 


Chronometer 

Sextant 

Chronometer 

Sun's 

Sun  East 

reading 

Sun  West 

limb 

22*    5m 

4' 

66° 

46' 

0" 

5*42™ 

21- 

Upper 

5 

48 

67 

o 

0 

41 

37 

u 

6 

33 

67 

18 

0 

40 

53 

« 

7 

17 

67 

34 

0 

40 

9 

« 

8 

1 

67 

50 

0 

39 

25 

u 

9 

8.5 

67- 

10 

0 

38 

17 

Lower 

9 

53 

67 

26 

0 

37 

32 

« 

10 

38 

67 

42 

0 

36 

47.5 

tt 

11 

23 

67 

58 

0 

36 

3 

tt 

22   12 

7.5 

68 

14 

0 

5  35 

18.5 

u 

6'  22     8 

35.3 

0"  5  38 

50.3 

104 


PRACTICAL   ASTRONOMY 


Reduction 
Mean  interval  t 


Amer.  Ephem.,  8 

48".7  x  3.742  =  rf8 

Sun's  apparent  a 

H0'  +  0") 

dt 
A0 


3  44 

3*.742 
56°  8' 
42  17 
13  16 

182".2 

2»Hm 

1  53 

+  18 


7«.5 
36.9 
30.6 


tan  <£ 
sin  t 

(1) 
tan  8 
tan  t 

(2) 


39«.3 

42.8 

11.4 

7.9 


log  ^8 


logd* 


9.9587 

9.9192 

1.095 

9.3725 

0.1732 

0.158 

9.9717 

2.2605 

8.8239 

1.0561 


79.  It  may  be  convenient  to  observe  the  equal  altitudes 
in  the  afternoon  of  one  day  and  the  forenoon  of  the  next 
day.  In  this  case  the  mean  of  the  two  observed  times 
minus  the  proper  value  of  dt  is  the  chronometer  time  of 
the  sun's  lower  culmination.  If  t  is  half  the  mean  time 
interval  between  the  observations  it  must  be  replaced  by 
180  +•  t  =  t'  when  substituting  in  (179)  ;  and  E  in  (180) 
and  a  in  (181)  must  be  increased  by  12A.  The  chronom- 
eter correction  at  midnight  is  then  given  by  (180)  and 
(181). 

Example.  Find  the  (sidereal)  chronometer  correction 
from  the  following  equal  altitude  observations  of  the  sun. 

jQ'    5*  33™  56«.3  Friday  afternoon,     1891  April  24 
0"  22     8   35  .3  Saturday  forenoon,  1891  April  24 


\  (0"  -O')=t 

Qh  17m  19«.5 

Reduction 

-1    21.5 

Mean  interval  t 

8  15    58  .0 

t 

8*.266 

t 

123°  59' 

t' 

303  59 

4> 

42  17 

8 

+  13     6 

+  49".0  x  8.266  =  rf8 

+  405".0 

12*  +  a 

14*   9M6«.4 

\  (0'  +  0") 

13  51    15  .8 

dt 

-25.4 

A0 

+  18     5  .2 

tan  <f> 

9.9587 

sin  t' 

9.9187n 

(1) 

-  1.096 

tan  8 

9.3668 

tan*' 

0.1713n 

(2) 

-  0.157 

-  (2)] 

9.9727,, 

log</8 

2.6075 

log^ 

8.8239 

log  eft 

1.4041n 

DETERMINATION    OF  .TIME  105 

dt  is  a  solar  interval,  and  should  be  reduced  to  sidereal, 
but  the  correction  is  small,  and  for  sextant  work  may  be 
neglected. 

80.  The  method  of  determining  time  by  equal  altitudes 
possesses  the  advantages  that  no  corrections  are  applied 
for  index  error,  eccentricity,  refraction,  parallax  and  semi- 
diameter;  any  undetermined  errors  are  eliminated  from 
the  result ;  and  the  latitude  need  not  be  accurately  known. 
However,  if  the  state  of  the  atmosphere  and  the  index 
correction  are  different  at  the  two  times  of  observation, 
the  equal  sextant  readings  do  not  correspond  to  equal 
true  altitudes,  and  a  correction  must  be  applied.  If  the 
index  correction  is  greater  and  the  refraction  less  for  the 
west  observation  than  for  the  east,  the  true  double  altitude 
at  the  west  observation  is  too  great  by  the  difference  of 
the  index  corrections  and  twice  the  difference  of  the  re- 
fractions, and  the  time  of  the  observation  must  be  increased 
by  the  interval  required  for  the  sextant  reading  to  decrease 
that  amount.  This  interval  can  be  determined  from  the 
observations  themselves.  Thus  in  the  example  of  §  78, 
the  index  correction  and  refraction  for  the  east  observation 

were 

/'  =  +  3'    8",     r'  =  1'  24" ; 

and  for  the  west 

I"  =  +  3'  21",     r"  =  V  22". 

The  true  double  altitude  at  the  west  observation  was  too 

great  by 

(/"  _  /')  +  2  (r'  -  r' )  =  +  17". 

From  the  observations  it  is  seen  that  the  sextant  reading 
decreased  16'  =  960"  in  about  44s.  If  x  is  the  correction 
to  the  time  of  the  west  observation,  we  have 

960  : 17  =  44  :  x, 

from  which  x= 0s. S.  The  correction  to  A0  is  —\x=  —  05.4, 
and  therefore  the  true  value  of  the  chronometer  correction 
is  A0  =  -f  18m  7'.5. 


106  PRACTICAL    ASTRONOMY 

81.  By  a  single  altitude  of  a  star.  A  series  of  double 
altitudes  of  a  star  having  been  observed  in  quick  succes- 
sion, let 

R  =  the  mean  of  the  sextant  readings, 
0'  =  the  mean  of  the  corresponding  chronometer  times ; 
and  let 

/  =  the  index  correction, 

e  =  the  correction  for  eccentricity, 
h'  =  the  apparent  altitude  of  the  star, 
z'  =  the  apparent  zenith  distance  of  the  star, 

r  =  the  refraction, 

z  —  the  true  zenith  distance  of  the  star. 

Then 

2  h'  =  R  +  I  +  t  =  2(90°  -2'),  (182) 

and 

z  =  z>  +  r.  (188) 

The  latitude  <f>  and  the  declination  8  being  known,  the 
hour  angle  t  is  given  by  (38)  or  (39).  The  sidereal  time 
at  the  instant  of  observation  is  given  by  0  =  a  -j-  t,  and 
thence  the  chronometer  correction  by 

A0  =  0  -  0'.  (184) 

In  case  a  mean  time  chronometer  is  used,  the  sidereal 
time  0  must  be  converted  into  the  mean  time  T  and  com- 
pared with  the  chronometer  time  T' . 

In  determining  the  time  from  single  altitudes  of  the  stars 
and  the  sun,  the  observations  should  not  be  confined  to  one 
side  of  the  meridian.  It  would  be  well  to  observe  alter- 
nately east  and  west  of  the  meridian,  at  about  equal  alti- 
tudes. A  comparison  of  the  results  of  such  a  series  often 
leads  to  the  detection  of  systematic  errors  whose  presence 
would  not  be  suspected  from  observations  made  wholly  in 
one  part  of  the  sky. 

Example.  Aldebaran  was  observed  east  of  the  meridian 
on  Wednesday  night,  1906  September  26,  at  Lick  Obser- 
vatory, with  the  following  mean  results  : 

0'=0»43™9«.2,        R  =  72°  12'  15",        1=  -4'  35",    e  =  0". 


DETERMINATION   OF   TIME 

Find  the  chronometer  correction. 


107 


R 

72°  12'  15" 

I 

4  35 

€ 

0 

i 

*7/ 

72     7  40 

h' 

36     3  50 

z' 

53  56  10 

From  (94) 

,  r 

z 

1    7 
53  57  17 

A> 

37  20  26 

Ephem.,  p.  338,  8 
<f>-8 

+  16  19  16 
21     1  10 

l-(t- 

3) 

3) 

74  58  27 
32  56    7 

Barom. 

Att.  Therm. 

Ext.  Therm. 

sin  Jj>  +  (<f>  -  8)] 

sin  i  0  -(<#»-  8)] 

sec  <f> 

sec  8 

From  (39),  sin2  \  t 

t 
t 
a 
0 
& 
A0 


25.8  inches 
59.5  F. 
60.0  F. 

9.78432 

9.45252 

0.09961 

0.01786 

9.35431 
151°  36'  27" 
303   12  54 
20*  12™  5K6 

4  30  33  .7 

0  43  25  .3 

0  43     9  .2 
+  16.1 


82.    By  a  single  altitude  of  the  sun.     If 

p  =  the  parallax  of  the  sun, 

S  =  the  semidiameter  of  the  sun, 

the  true  zenith  distance  of  the  center  of  the  sun  is  given  by 

z  =  z'  +  r-p±S;  (185) 

S  being  +  or  —  according  as  the  upper  or  lower  limb  of 
the  sun  was  observed.  The  value  of  t  is  given  by  (38)  or 
(39)  as  before,  t  is  the  true  time  when  the  observation 
was  made.  The  mean  time  T  is  given  by  applying  the 
equation  of  time  E.  If  T'  is  the  chronometer  time  of 
observation,  the  chronometer  correction  is 

&T=T-T.  (186) 

If  a  sidereal  time-piece  is  used  the  mean  time  T  must  be 
converted  into  the  sidereal  time  and  the  resulting  value 
compared  with  the  chronometer  time. 

Since  the  declination  of  the  sun  is  changing,  it  is  neces- 
sary to  know  the  chronometer  correction  within  10s;  other- 
wise the  value  of  B  taken  from  the  Ephemeris  may  be 


108 


PRACTICAL   ASTRONOMY 


slightly  in  error,  thus  giving  only  an  approximate  value 
of  the  chronometer  correction.  With  this  value  of  the 
chronometer  correction  a  more  accurate  value  of  $  could  be 
found,  which  substituted  in  (38)  as  before  would  give  prac- 
tically exact  values  of  t  and  the  chronometer  correction. 

Example.    The  observations  made  on  the  sun  east  of  the 
meridian,  recorded  in  §  78,  give 

0'  =  22*  8m  3S-.3,  R  =  67°  30'  0". 

The  chronometer  correction  is  assumed  to  be  -f  18m35; 
required  its  value  furnished  by  the  observations. 

Barom.  29.036  inches 
Att.  Therm.  50°.0  F. 
Ext.  Therm.  47  .8  F. 

Amer.  Ephem.,  p.  278,  ir  S".8 

log  tr  0.944 

sin  z'  9.920 

From  (64),  jo  7" 

B  22*  8"  35* 

Approx.  A0  + 18     3 

Sid.  time  22  26   38 

Mean  time  April  24<*20  13  29 
Longitude  5  34  55 

Gr.  mean  time  April  25    1  48  24 
Ephem.,  8  +  13°  12'  53" 

True  time  April  24d20A15"*38«.3 
Longitude  5  34  55  .1 

Gr.  true  time  April  25     1  51 
Eq.  of  time,  E  -  2     4.9 

Mean  time  April  24  20  13   33  .4 
Sid.  time,  0  22  26  42  .4 

Chron.time,0'  22    8  35.3 

A0  -|- 18     7 .1 


This  value  of  A0  differs  so  little  from  the  assumed  value 
that  another  approximation  to  the  value  of  B  is  unnecessary. 


R 

67°  30'    0" 

I 

+  3     8 

€ 

12 

2hf 

67  32  56 

h' 

33  46  28 

z' 

56  13  32 

r 

1  24 

P 

7 

z 

56  14  49 

<£ 

42  16  47 

8 

+  13  12  53 

<£  -  8 

29     3  54 

<£  +  8 

55  29  40 

z  4.  (<£  _  g) 

85  18  43 

2  -(<£-  8) 

27  10  55 

Z  +  (<£  +  8) 

111  44  29 

z  —  (<£  +  g) 

0  45     9 

sin  £  [z  +  (<£  —  8)] 

9.83097 

sin  £  [z  -  (<}>  -  8)] 

9.37104 

sec  %  [z  4-  (<j!>  +  8)] 

0.25099 

sec  i  [2  -(<#>  +  8)] 

0.00001 

tan2  i  < 

9.45301 

tan£f 

9.72650n 

H 

151°  57'  17" 

< 

303  54  34 

True  time 

20*15"»38'.3 

GEOGRAPHICAL  LATITUDE  109 

Using  +  3'  21"  as  the  index  correction,  the  value  of 
A#  given  by  the  afternoon  solar  observations,  §  78,  is 
+  18m  85.1,  which  agrees  well  with  the  above,  assuming  the 
chronometer's  daily  rate  to  be  -f  35.6  [§  64]. 

83.  The  error  in  the  hour  angle  —  and  therefore  in  the 
time  —  produced  by  a  small  error  in  the  measured  altitude 
or  in  the  assumed  latitude  is  readily  found.  Differentiat- 
ing (15),  regarding  z  and  t  as  variables,  and  reducing  by 
(17),  we  obtain 

dt=  .     d{z        ;  (187) 

sin  A  cos  <j> 

that  is,  an  error  dz  in  the  measured  zenith  distance  pro- 
duces an  error  dt  in  the  time,  which  is  least  when  sin  A  is 
a  maximum.  Likewise,  differentiating  (15)  with  respect 
to  </>  and  t  and  reducing  by  (16)  and  (17),  we  have 


tan  A  cos 


(188) 


that  is,  an  error  d$  in  the  latitude  gives  rise  to  an  error 
dt  in  the  time,  which  is  small  when  tan  A  is  large. 

For  these  reasons  it  appears  that  to  obtain  the  best 
determination  of  time  from  observed  altitudes,  those  stars 
should  be  selected  which  are  as  nearly  as  possible  in  the 
prime  vertical. 

GEOGRAPHICAL   LATITUDE 

84.  By  a  meridian  altitude  of  a  star  or  the  sun.  Observe 
the  double  altitude  of  the  star  or  sun  at  the  instant  when 
it  is  on  the  meridian,  and  obtain  the  true  zenith  distance  z 
as  in  §§81  and  82.  The  latitude  is  then  found  from 

4>  =  S±z,  (189) 

the  upper  sign  being  used  for  a  star  south  of  the  zenith, 
the  lower  sign  for  a  star  between  the  zenith  and  the  pole. 
For  a  star  below  the  pole  we  have 

<£  =  180°  -  8  -  z.  (190) 


110  PRACTICAL   ASTRONOMY 

Example.  The  double  altitude  of  the  sun's  lower  limb 
was  observed  at  Ann  Arbor  at  true  noon,  Friday,  1891 
Feb.  6,  as  follows  : 

Sextant  63°  49'  15".     Barom.  28.98  inches,     Ext.  Therm.  38°  F. 

Find  the  latitude. 

In  this  case  r  is  computed  from  (96),  and  p  may  be  taken 
from  the  table  on  page  27. 

R     63°  49'  15"  z'  58°    3'  56" 

7+35  r  1  28 

c      -        12  p                     8 

2h'    63  52     8  S  -  16  15 

A'    31   56     4  z  57  49     1 

z'    58     3  56  8  - 15   32  11 

<f>  42   16  50 

85.  By  an  altitude  of  a  star,  the  time  being  known.  Hav- 
ing determined  the  star's  hour  angle  by  (41),  the  latitude 
is  given  by  (15),  in  which  <£  is  the  only  unknown  quantity. 
To  determine  it,  assume 

/  sin  F  =  cos  8  cos  t,  (191) 

/cosF=sin8,  (192) 

and  (15)  becomes 

cos  z  =/sin  (<£  +  F)  =  sin  8  sec  Fsin  (<J>  +  F). 

From  these  we  obtain 

tan  F  =  cot  8  cos  f,  (193) 

sin  (<j>  +  F)  =  cos  F  cos  z  cosec  8,  (194) 

which  effect  the  solution. 

The  quadrant  of  F  is  determined  by  (191)  and  (192). 
(<£  -f-  F),  being  determined  from  its  sine,  may  terminate 
in  either  of  two  quadrants,  thus  giving  rise  to  two  values 
of  the  latitude.  That  one  is  selected  which  agrees  best 
with  the  known  approximate  value  of  the  latitude. 

In  case  the  sun  is  observed,  t  is  the  true  solar  time. 


GEOGRAPHICAL   LATITUDE 


111 


Example.     Find  the  latitude  from  the  following  double 
altitudes  of  Polaris  observed  Saturday,  1891  April  25 : 


Chronometer 
14*  55™    5« 
56    35 

Sextant 
82°  18'  40" 
19  20 

Barom. 
Ext.  Therm. 

29.17  inches 
39°.3  F. 

58 

20 

19 

35 

Amer.  Ephem.,  p.  305 

15 

1 

0 

20 

40 

a       1» 

17M8» 

Means 

14 

57 

45 

82 

19 

34 

8    88°  43'  32" 

0' 

14*  57"  45* 

R 

82°  19'  31" 

cot  8 

8.347270 

A0 

+ 

18 

10 

I 

+  3 

12 

cost 

9.939572* 

0 

15 

15 

55 

c 

15 

F 

358°  53'  28" 

a 

1 

17 

48 

2  A' 

82 

22 

31 

cos  F 

9.999919 

t 

13 

58 

7 

hr 

41 

11 

15 

cosz 

9.818414 

t 

209° 

31' 

45" 

z' 

48 

48 

45 

cosec  8 

0.000108 

r 

1 

6 

sin  (<£  +  F) 

9.818441 

z 

48 

49 

51 

<f>  +  F 

41°  10'  20" 

6 

42   16  52 

86.   Differentiating  (15)  with  regard  to  z  and  <f>  and 
reducing  by  (16)  we  obtain 

1 1         dz 


cos  A ' 


(195) 


that  is,  an  error  dz  in  the  measured  zenith  distance  pro- 
duces the  minimum  error  d(f>  in  the  latitude  when  the  star 
is  on  the  meridian. 

Differentiating  (15)  with  regard  to  $  and  £,  and  reduc- 
ing by  (16)  and  (17),  we  obtain 


d<f>  =  —  tan  A  cos  <£  dt ; 


(196) 


that  is,  an  error  dt  in  the  estimated  time  of  making  the 
observation  gives  rise  to  an  error  d<j>  in  the  latitude,  which 
wili  be  small  when  the  star  is  nearly  on  the  meridian,  and 
equal  to  zero  when  A  is  0°  or  180°. 

For  these  reasons  it  appears  that  to  obtain  the  best  de- 
termination of  the  latitude  from  observed  altitudes,  those 
stars  should  be  selected  which  are  as  nearly  as  possible  on 
the  meridian. 


112  PRACTICAL  ASTRONOMY 

87.  By  circummeridian  altitudes.  The  method  of  §  84 
is  applicable  to  only  one  altitude  observed  when  the  star 
is  on  the  meridian.  If  a  series  of  altitudes  be  observed 
just  before  and  after  meridian  passage,  —  called  circum- 
meridian  altitudes,  —  they  can  be  reduced  to  the  equivalent 
meridian  altitudes  and  a  quite  accurate  value  of  the  lati- 
tude obtained  by  combining  the  results.  Equation  (15) 
may  be  written 

cos  z  =  cos(<£  —  8)  —  cos  <f>  cos  8  2  sin2  \  t.  (197) 

If  we  let  20  be  the  zenith  distance  of  the  star  when  it  is  on 
the  meridian  and  put  y  =  cos  <f>  cos  S  2  sin2  J  £,  (197)  be- 
comes 

cos  z  =  cos  z0  —  y.  (198) 

Here  z  is  a  function  of  j/,  and  we  may  write 

*=/(?)• 

Developing  this  in  series  by  Maclaurin's  formula,  restoring 
the  value  of  ?/,  and  dividing  the  abstract  terms  by  sin  1"  to 
express  them  in  seconds  of  arc,  we  have 

cos  <£  cos  8    2  sin2 1  £     /cos<£cosSy   cot  ZQ  2  sin4 \ 1 
=  ZQ  +      sinz0  sTn~T'r  ~(     sinz0      )  '    ~lfal»~ 

which  converges  rapidly  when  t  does  not  exceed  30m,  and 
the  star  is  more  than  20°  from  the  zenith,  as  it  will  be  in 
sextant  double  altitudes.     If  we  let 
cos  <j>  cos  8 


smz,, 


=  A,          A*cotz0  =  B,  (200) 


,nM. 

.  =  m,  .      *    =  n,  (201) 

sin  1"  sin  I" 

and  substitute  the  resulting  value  of  ZQ  for  z  in  (189),  we 
have 

<£  =8±z^Am±  Bn,  (202) 

the   lower   sign   being   employed   for  a  star  culminating 
between  the  zenith  and  the  pole. 

When  a  star  is  observed  near  the  meridian  at  lower  cul- 
mination, it  is  convenient  to  reckon  the  hour  angle  from 


GEOGRAPHICAL   LATITUDE  113 

the  lower  transit,     t  in  (15)  must  be  replaced  by  180°  4-  t, 
and  we  obtain 

cos  z  =  sin  <£  sin  8  —  cos  <£  cos  8  cos  t  =  —  cos  (<j>  -f  8)  +  cos  <f>  cos  8  2  sin2  £  £. 

Developing  this  in  series  as  before  and  substituting  the 
resulting  value  of  z0  for  z  in  (190),  we  have 

<j>  =  180°  -  8  -  z  -  Am  -  Bn.  (203) 

The  entire  series  of  observations  is  conveniently  reduced 
as  a  single  observation  by  letting  2,  m  and  n  in  (202)  and 
(203)  represent  the  arithmetical  means  of  the  values  of 
these  quantities  for  the  individual  observations. 

The  values  of  m  and  n  are  tabulated  in  the  Appendix, 
TABLE  III,  with  the  argument  t. 

An  approximate  value  of  <£  is  required  in  computing  A. 
This  may  be  obtained  by  the  method  of  §  84,  from  the 
observation  made  nearest  the  meridian. 

If  the  sun  is  observed,  the  declination  is  taken  from  the 
Ephemeris  for  the  instant  of  each  observation  in  case  the 
observations  are  reduced  separately,  and  for  the  mean  of 
the  times  in  case  they  are  reduced  collectively. 

If  a  star  is  observed  with  a  sidereal  chronometer,  the 
hour  angles  t  are  the  intervals  between  each  observed  time 
and  the  chronometer  time  of  the  star's  transit. 

If  a  star  is  observed  with  a  mean  time  chronometer,  the 
intervals  must  be  reduced  from  mean  to  sidereal  intervals 
before  entering  TABLE  III  for  m  and  n. 

If  the  sun  is  observed  with  a  mean  time  chronometer,  the 
intervals  should  be  reduced  to  apparent  solar  intervals  by 
correcting  for  the  change  in  the  equation  of  time  during 
the  intervals.  This,  however,  will  never  exceed  05.5,  and 
may  be  neglected  in  sextant  observations. 

If  the  sun  is  observed  with  a  sidereal  chronometer,  the 
intervals  must  be  reduced  to  mean  solar  intervals  and 
thence  to  apparent  solar. 

If  the  rate  of  the  chronometer  is  large  it  must  be  allowed 
for. 


114 


PRACTICAL   ASTRONOMY 


Example.  Wednesday,  1891  April  8,  at  a  place  in  lati- 
tude about  42°  17'  and  longitude  5"34W55*  the  following 
double  altitudes  of  the  sun  were  observed  with  a  sextant 
and  sidereal  chronometer.  Barom.  29.373  inches,  Att. 
Therm.  66°  F.,  Ext.  Therm.  42°.o  F.  Required  the  lati- 
tude. [Each  printed  observation  is  the  mean  of  three 
consecutive  original  observations.] 


Limb 

Sextant 

Chronometer 

Sidereal  t 

Solar  t 

m 

n 

Upper 

109°  58'  33".7 

0*  31"  28*.0 

—  19™  59«.2 

—  19"*  55^.9 

77<)".6 

1".47 

Lower 

109     4  40  .0 

34    51.0 

—  16    36.2 

—  16     33.5 

538  .1 

0  .70 

« 

109   14     6  .2 

38    43.3 

—  12    43.9 

—  12    41.8 

316  .4 

0  .24 

Upper 

110   25  16  .2 

42    42.3 

—   8    44.9 

-   8    43.5 

149  .5 

0  .05 

" 

110   29  39  .0 

45    44.7 

-   5    42.5 

—   5    41.6 

63  .6 

0  .01 

Lower 

109   27  27  .5 

49    30.7 

—   1    56.5 

—   1    56.2 

7  .4 

0  .00 

" 

109   27  43  .7 

53      9.7 

+   1    42.5 

+   1    42.2 

5  .7 

0  .00 

Upper 

110   29    0  .0 

0  57    36.0 

+  6      8.8 

+   6      7.8 

73  .8 

0  .01 

" 

110   21  51  .2 

1     2    56.0 

+  11    28.8 

+  11    26.9 

257  .3 

0  .16 

Lower 

109   11  33  .7 

5    46.7 

+  14    19.5 

+  14    17.2 

400  .6 

0  .39 

11 

109     2  27   .7 

9    12.3 

+  17    45.1 

+  17    42.2 

615  .0 

0  .92 

Upper 

109   56  20  .0 

1    12    33.7 

+  21      6  .5 

+  21      3  .0 

869  .4 

1  .83 

109  45  43  .2 


+  0    33.9     339  .7      0  .48 


Apparent  time  of  apparent  noon  0*   Qm   08.0 

Equation  of  time  +   1     52.1 

Mean  time  of  apparent  noon  0     1     52.1 

Sidereal  time  of  apparent  noon  1     8     37.2 

Chronometer  correction  +  17     10.0 

Chronometer  time  of  apparent  noon  0  51     27.2 

The  difference  between  this  and  the  observed  times  gives 
the  sidereal  intervals  t  as  above. 

The  mean  of  the  hour  angles  is  +  Om  335.9,  and  there- 
fore the  sun's  declination  is  taken  for  the  local  mean  time 
0*  1™  52M  +  0™  335.9  =  0"  2™  26s,  or  Greenwich  mean  time 
5*  37™  215. 

An  equal  number  of  observations  on  the  upper  and 
lower  limbs  was  made,  hence  there  is  no  correction  for 
semidiameter. 

The  solution  of  (202)  is  made  as  follows : 


GEOGRAPHICAL   LONGITUDE 

8+7°  11'  33".4                       Sextant  109°  45'  43".2 

<£  42   17  /  +2  51  .7 

z0  34  59  26  .6  *  18  .0 

cos<£        9.86913  2A'  109   48  16  .9 

cos  8       9.99647  W  54   54     8  .4 

cosecz0       0.24151  «'  35     5  51  .6 

log  ,4        0.10711  r'  40  .5 

logm        2.53110  P  5  .1 

Am       434".7  «  35     6  27  .0 

log ,42       0.2142  8  +7   17  33  .4 

cotz0       0.1549  Am  7  14  .7 

logw        9.6812  Bn  1  .1 

log£n        0.0503  <f>  42   16  46  .8 

Bn  I'M 

A  repetition  of  the  computation  with  this  value  of  $  does 
not  change  the  result. 

[The  latitude  of  the  place  is  known  to  be  about 
42°  16'  47".!.] 

GEOGRAPHICAL  LONGITUDE 

88.  By  lunar  distances.     The  moon's  distance   from  a 
star  nearly  in  the  ecliptic  is  rapidly  changing.     Its  geo- 
centric distances  from  the  sun,  Venus,  Mars,  Jupiter,  Saturn 
and  nine  bright  stars  near  its  path  are  given  in  the  Ameri- 
can Ephemeris  [pp.  XIII-XVIII  of  each  month]  at  three- 
hour  intervals  of  Greenwich  mean  time,  from  which  the 
distances  at  any  other  instants  may  be  found  by  interpola- 
tion.    Conversely,  if  its  distance  from  any  of  these  objects 
is  measured  with  the  sextant  and  the  apparent  distance 
reduced    to    the    corresponding   geocentric   distance,   the 
Greenwich  mean  time  at  the  instant  of  observation  may 
be  found.    The  Greenwich  mean  time  minus  the  observer's 
mean  time  is  the  observer's  longitude.     This  method  of 
determining  the  longitude  is  occasionally  of  considerable 
importance  to  navigators  and  explorers. 

89.  We  shall  suppose  that  the  moon's  distance  from  the 
sun  has  been  observed.     The  formulse  for  a  planet  will  be 
the  same,  save  that  the  semidiameter  of  the  planet  may 


116  PRACTICAL   ASTRONOMY 

usually  *  be  neglected.     For  a  star  the  parallax  and  semi- 
diameter  are  zero. 

The  sextant  reading  having  been  corrected  for  the  index 
error  and  eccentricity,  the  result  is  the  apparent  distance 
between  the  nearest  limbs  of  the  sun  and  moon.  It 
must  be  corrected  for  their  semidiameters,  refractions  and 
parallaxes. 

To  compute  these  corrections,  the  zenith  distances  of 
the  two  bodies  must  be  known.  When  there  are  three 
observers,  as  frequently  happens  at  sea,  the  altitudes  of  the 
sun  and  moon,  and  the  distance  between  them,  should  be 
measured  simultaneously.  The  observer's  mean  time  can 
be  obtained  also  from  these  observed  altitudes  of  the  sun 
[§  82].  When  it  is  not  practicable  to  make  these  obser- 
vations at  the  same  time,  the  observer  may  measure  the 
altitudes  immediately  before  and  after  measuring  the  lunar 
distance,  and  obtain  the  required  altitudes  at  the  instant 
of  observation  by  interpolation.  Again,  the  observer  may 
assume  an  approximate  value  of  the  longitude  (which  he 
can  usually  do  sufficiently  accurately),  and  take  from  the 
Ephemeris  the  right  ascensions  and  declinations  of  the 
sun  and  moon  corresponding  to  the  Greenwich  time  thus 
obtained.  The  hour  angles,  azimuths  and  zenith  distances 

are  then  given  by  §§  8  and  5. 

The  parallax  of  the  sun  in  azi- 
muth is  negligible ;  its  parallax 
in  zenith  disjtance  is  given  by 
(64).  The  parallax  of  the  moon 
in  azimuth  is  given  by  (71)  and 
(72) ;  and  in  zenith  distance,  by 
(80),  (78)  and  (79).  (95)  gives 
the  refractions,  care  being  taken  to 
use  the  apparent  zenith  distance. 
FIG.  18  The  semidiameters  of  the  sun  and 

*  In  case  the  telescope  is  powerful  enough  to  define  the  planet's  disk, 
the  moon's  limb  may  be  made  to  pass  through  the  center  of  the  disk. 


GEOGRAPHICAL   LONGITUDE  117 

moon  are  obtained  by  the  methods  of  §§  33-35.  The  solu- 
tion of  (107)  requires  the  values  of  q.  In  Fig.  18  let  M 
be  the  moon's  center,  S  the  sun's  center,  and  Z  the  zenith. 
For  the  sun,  q  =  ZSM,  and  for  the  moon,  q  —  ZMS.  If 
we  let 

Z1  =  the  apparent  zenith  distance  of  the  sun  =  ZS, 
zf  =  the  apparent  zenith  distance  of  the  moon  =  ZM, 
df  =  the  apparent  distance  between  the  centers  =  SM, 

we  can  write,  for  the  sun, 


sin  I  (zf  -Z'  +  d'}  sin  j  (z'  +  Z'  -  d') 
tan i ?  =\8in  4  (Z,  +  a,  +  dl)  sin  i  (Z,  _z,+  dl)  »        (204) 

and  for  the  moon, 


sini(*'  +  ^'  +  O  sinks'  -  Z'  +  d') 

Adding  the  inclined  semidiameters  given  by  (107)  to 
the  corrected  sextant  reading,  the  sum  is  the  distance  d' 
between  the  centers  as  seen  from  the  observer. 

The  combined  effect  of  the  refraction  and  the  parallax 
in  zenith  distance  is  to  shift  the  bodies  in  their  vertical 
circles  without  changing  the  angle  SZM  at  the  zenith, 
which  we  shall  represent  by  V.  If  we  let 

Z  =  the  geocentric  zenith  distance  of  the  sun, 
z  =  the  geocentric  zenith  distance  of  the  moon, 
d"  =  the  corresponding  distance  between  the  centers, 

we  can  write 

cos  d"  =  cos  z  cos  Z  +  sin  z  sin  Z  cos  V,  (206) 

cos  d'  =  cos  z'  cos  Z'  +  sin  z'  sin  Z1  cos  V.  (207) 

Therefore 

cos  d"  —  cos  z  cos  Z  _  cos  d'  —  cos  z'  cos  Z' . 

sin  z  sin  Z  sin  z'  sin  Z' 

or 

cos  d"  -  cos  (z  4-  Z)  _  cosd'  —  cos  (z'  +  Z')  (208^ 

sin  2  sin  Z  sinz'sinZ' 


118  PRACTICAL   ASTRONOMY 

If  we  put  z'  +  Zf  +  d'  =  2  x,  and  substitute 

cos  d'  —  cos  (z'  -f  Z')  =  2  sin  x  sin  (a;  —  d'), 

cos  d"  =  1  -2  sin2  1  d", 
cos  (2  -f  Z)  =  2cos2i  (2  +  Z)  -  1, 
=  1  -  2  sin2  i  (2  +  Z), 
(208)  reduces  to 

sin2  1  d"  =  sin2  $  (2  +  Z  )  -  g|°!,^"f,  sin  *  sin  (*  -  <*')•     (209) 

Let  an  auxiliary  angle  M  be  defined  by 
sin  z  sin  Z      g 


sin  2'  sin  Z'        sin2  £  (2  +  Z) 

Then  (209)  takes  the  form 

sin  i  d"  =  sin  £  (2  +  Z)  cos  If  .  (211) 

The  parallax  of  the  moon  in  azimuth  produces  a  small 
change  in  V  and  therefore  in  d".  From  (206),  by  differ- 

entiation, 

M"  =  sin  z  sin  Z  sin  F  cosec  d"  A  F,  (212) 

in  which  AJ^is  the  parallax  in  azimuth. 

The  geocentric  distance  d  between  the  centers  is  now 

given  by 

d  =  d"  +  Ad".  (213) 

In  connection  with  the  lunar  distances,  the  Ephemeris 
gives  a  column  "  P.  L.  of  Diff."  (Proportional  Logarithm 
of  the  Difference),  which  is  the  logarithm  of  10800,  the 
number  of  seconds  in  3*,  minus  the  logarithm  of  the  change 
in  the  lunar  distance,  expressed  in  seconds  of  arc,  in  the 
next  following  three  hours.  That  is,  it  is  the  logarithm  of 
the  reciprocal  of  the  moon's  average  rate  for  the  three 
hours,  or  the  rate  at  the  middle  period  of  the  three  hours 
[see  remarks  on  interpolation,  §15].  In  order  to  interpolate 
for  the  Greenwich  mean  time  corresponding  to  the  given 
value  of  c?,  we  have  only  to  add  the  P.  L.  of  Diff.  for  the 
middle  period  of  the  approximate  interval  to  the  logarithm 
of  the  number  of  seconds  of  arc  by  which  d  exceeds  the 


GEOGRAPHICAL   LONGITUDE  119 

next  smaller  Ephemeris  lunar  distance.  The  sum  is  the 
logarithm  of  the  number  of  seconds  of  time  by  which 
the  Ephemeris  time  is  to  be  increased. 

If  the  P.  L.  of  Diff.  given  in  the  Ephemeris  is  used 
without  change,  a  slight  correction  for  the  neglected  second 
difference  of  the  moon's  rate  can  be  taken  from  TABLE  I, 
Appendix,  American  Ephemeris,  and  applied  as  there 
directed. 

If  the  resulting  longitude  differs  considerably  from  the 
assumed  longitude,  a  second  approximation  should  be  made 
by  starting  with  the  value  of  the  longitude  just  obtained. 
A  third  approximation  will  not  be  necessary. 

Example.  Tuesday,  1891  May  12,  the  distance  between 
the  bright  limbs  of  the  sun  and  moon  was  observed  with 
a  sextant  and  sidereal  chronometer.  The  mean  of  ten 
observations  gave 

0'  =  8*  36™  10s,     R  =  57°  28'  32". 9. 

Chronometer  correction,  +19"1155;  index  correction, 
+  2'  56".4;  Barom.  29.25  inches,  Att.  Therm.  62°  F., 
Ext.  Therm.  57°  F. ;  latitude,  +  42°  16'  47" ;  longitude 
assumed,  4-  5A  34m.  Required  a  more  exact  value  of  the 
longitude. 

0'      SA36TO10*  R    57°28'32".9 

A0      +  19    15  /     +     2  56  .4 

0      8  55   25  €     -         11  .3 

Mean  time      5  33   42  Distance    57  31  18  .0- 

Longitude      5  34 
Gr.  mean  time     11     7    42 

Corresponding  to  this  Greenwich  mean  time,  we  take 
from  the  American  Ephemeris,  pp.  74,  75,  77,  80  and  278, 

Sun  Moon 

Right  ascension,         a        3*  17W  54«  7*  29W  31« 

Declination,                8  +  18°  15'  10"  +  25°  36'  55" 

Semidiameter,            S            15   51.7  15   12.8 

Horizontal  parallax,  TT                     8.8  55   43.3 


120  PRACTICAL   ASTRONOMY 

By  §§  8  and  5  we  find  for  the  geocentric  coordinates  of 
the  sun, 

t  =  84°  22'  45",        A  =  100°  8'  50",        Z  =  73°  46'  5" ; 
and  for  the  moon, 

t  =  21°  28'  30",        A  =  53°  27'  21",         z  =  24°  15'  40". 
Computing  the  parallaxes  we  obtain,  for  the  sun, 
A'-A  =  0,       p  =  Z'  -  Z  =  8".4. 

From  (58)  we  find  <f>  -  $  =  687".3  =  IV  27".3;  and  from 
(59)  log  p  =  9.99935 ;  therefore,  for  the  moon, 

log  m  =  6.11807,         A'  -  A  =  +  21".8, 
y  =  +  6'  49".2,         log  n  =  8.20908,        z'  -  z  =  23'  6".2. 

The  mean  refraction  of  the  sun,  TABLE  II,  is  about 
3' 13",  and  therefore  its  apparent  zenith  distance  is  very 
nearly  73°  43'  0".  The  value  of  the  refraction  is  now 
found  from  (95)  to  be  3'  8".6.  Similarly,  the  refrac- 
tion for  the  moon  is  25". 6.  The  apparent  zenith  distances 
of  the  sun  and  moon  are  therefore 

Z'  =  73°  43'  4".8,        z'  =  24°  38'  20".6. 

The  apparent  zenith  distance  of  the  upper  limb  of  the 
sun  is  73°43'4".8-15'51".7  =  73°27'13".l.  The  cor- 
responding refraction  is  3'  5". 5.  The  apparent  vertical 
semidiameter  is  therefore  contracted  3".l  [§  35],  and  its 
value  is  15'  48".6. 

The  moon's  apparent  semidiameter  is  found  from  (106) 
to  be  15'  26". 3 ;  and  by  refraction  its  apparent  vertical 
semidiameter  is  reduced  to  15'  26".0. 

The  approximate  distance  between  the  centers  of  the 
sun  and  moon  is 

d'  =  57°  31'  18"  +  15'  49"  +  15'  26"  =  58°  2'  33". 

Substituting  these  values  of  df,  Zf  and  z'  in  (204)  we 
obtain  for  the  sun,  q  =  20°  58' ;  and  in  (205)  for  the  moon, 
q  =  124°  34'.  For  the  sun,  a  =  15'  51".7  =  951".7,  b  = 
15'48".6  =  948".6;  and  by  (107)  the  inclined  semidiameter 


GEOGRAPHICAL  LONGITUDE  121 

is  #"  =  15'  49".l.  Similarly,  for  the  moon,  S"  =  15'  26".2. 
The  apparent  distance  between  the  centers  of  the  sun  and 
moon  is  therefore 

d'  =  57°  31'  18".0  +  15'  49".l  +  15'  26".2  =  58°  §'  33".3. 

The  solution  of  (210)  gives  M=  49°  48'  39".0  ;  and 
thence,  from  (211),  d"  =  58°  18'  16".0.  Substituting 
A'  -  A  =  AF=  +  21".8  in  (212),  we  obtain  Ad"=  +  7".4. 
The  geocentric  distance  between  the  sun  and  moon  is 
therefore,  by  (213), 

d  =  58°  18'  16".0  +  7".4  =  58°  18'  23"  A. 

From  the  American  Ephemeris,  pp.  86  and  87,  at 

Greenwich  mean  time    9*,  d  =  57°  16'  23",  P.  L.  of  Din0.  =  0.3169, 

Greenwich  mean  time  12  ,  d  =  58  43    9  ,          P.  L.  of  Diff.  =  0.3184. 

We  have  to  interpolate  for  the  interval  of  time  T  after  9*, 
corresponding  to  a  change  in  d  of  58°  18'  23".4-57°  16'  23" 
=  3720".4.  The  value  of  T  is  approximately  2*.  The 
value  of  P.  L.  of  Diff.  at  the  middle  of  the  2*  is  0.3167. 

P.  L.  of  Diff.  0.3167  Gr.  mean  time  11*   8™  34* 

log  3720".4  3.5706  Observer's  mean  time     5  33  42 

log  T  3.8873  Observer's  longitude     5  34  52 

T        7714' 
T 


The  true  value  of  the  longitude  is  known  to  be  5^  34m  55*. 
The  error  of  3s  corresponds  to  an  error  of  2"  in  the  meas- 
ured distance  [or  in  the  lunar  tables],  and  is  unusually 
small.  The  observations  are  difficult  to  make,  and  the 
measures  of  the  best  observers  are  easily  liable  to  an  error 
of  10".  It  is  well,  however,  to  carry  the  numerous  correc- 
tions to  tenths  of  a  second  to  prevent  the  accumulated 
effect  of  neglected  fractions. 

The  above  solution  of  this  problem  is  essentially  a 
rigorous  one.  Navigators  are  accustomed  to  employ 
abridged  forms  of  solution,  for  which  the  reductions  are 
much  shorter.  Likewise  many  of  the  functions  are  tabu- 
lated, which  still  further  reduces  the  labor. 


CHAPTER  VII 
THE   TRANSIT   INSTRUMENT 

90.  The  transit  instrument  consists  essentially  of  a 
telescope  attached  perpendicularly  to  a  horizontal  axis. 
The  cylindrical  extremities  of  this  axis  are  the  pivots. 
The  straight  line  passing  through  their  centers  is  the 
rotation  axis.  The  supports  for  the  two  pivots  are  called 
the  Vs.  The  straight  line  passing  through  the  optical 
center  of  the  object  glass  and  the  rotation  axis  and  per- 
pendicular to  the  latter  is  the  collimation  axis.  By  revolv- 
ing the  instrument  about  the  rotation  axis  the  collimation 
axis  describes  a  plane  called  the  collimation  plane.  In  the 
common  focus  of  the  object  glass  and  eye-piece  is  a  system 
of  wires  called  the  reticle.  It  consists  either  of  spider 
threads  attached  to  a  frame,  or  of  fine  lines  ruled  on  thin 
glass.  An  odd  number  of  wires  —  usually  five,  seven  or 
eleven — is  placed  parallel  to  the  collimation  plane  and 
perpendicular  to  the  collimation  axis,  over  which  the  times 
of  transit  of  a  star's  image  are  observed.  The  middle  wire 
of  the  set  is  fixed  as  nearly  as  possible  in  the  collimation 
plane.  One  or  two  wires  are  placed  perpendicular  to  these 
to  mark  the  center  of  the  field  of  view.  A  micrometer 
wire  parallel  to  the  first  set  is  arranged  to  move  as  nearly 
as  possible  in  their  plane.  The  axis  of  the  instrument  is 
hollow.  A  light  is  placed  so  that  the  rays  from  it  enter 
the  axis  and  fall  on  a  small  mirror  in  the  center  of  the 
telescope,  which  reflects  them  to  the  eye-piece  in  such  a 
way  that  the  wires  are  seen  as  dark  lines  in  a  bright  field. 
The  illuminating  apparatus  in  some  instruments  is  arranged 

122 


THE   TRANSIT   INSTRUMENT 


128 


so  that  the  observer  may  change  from  dark  lines  in  a  bright 
field  to  bright  lines  in  a  dark  field,  —  a  necessary  arrange- 
ment when  the  object  to  be  observed  is  very  faint.  A 

very  common  distribution  of  the 
wires  in  the  reticle  is  shown  in 
Fig.  19. 

The  instrument  is  so  arranged 
that  its  rotation  axis  can  be 
rotated  180°  ;  i.e.,  reversed,  about 
a  vertical  line.  The  two  posi- 
tions are  defined  conveniently  by 
stating  the  position  of  the  clamp 
or  graduated  circle  on  the  axis. 
Thus,  clamp  W  or  clamp  E,  circle  W  or  circle  E,  denotes 
that  position  of  the  instrument  in  which  the  clamp  or 
circle  is  west  or  east  of  the  collimation  plane. 

An  excellent  form  of  transit  and  zenith  telescope  com- 
bined is  shown  in  Fig.  20.  The  circular  base-plate  of  the 
instrument  is  supported  on  three  screws,  with  which  the 
instrument  may  be  quickly  leveled  on  its  supporting  pier. 
The  two  standards  which  support  the  pivots  of  the  instru- 
ment are  rigidly  fixed  to  a  circular  plate,  which  may  be 
rotated  freely  through  180°  on  the  base-plate  when  the 
instrument  is  used  as  a  zenith  telescope  [considered  in  the 
next  chapter].  When  the  instrument  is  used  as  a  transit, 
the  two  circular  plates  are  clamped  together,  and  remain 
clamped.  To  reverse  the  instrument,  the  observer  turns 
the  reversing  crank  (shown  in  the  lower  right  corner  of  the 
cut),  which  raises  the  two  small  inner  standards  until  the 
pivots^  are  entirely  free  from  the  V's ;  the  axis,  supported 
on  the  two  standards,  is  then  turned  gently  through  180°, 
and  carefully  lowered  by  turning  the  crank  in  the  reverse 
direction,  until  the  pivots  rest  in  the  V's.  The  illuminat- 
ing lanterns  are  in  position  for  the  rays  to  pass  through 
the  pivots.  The  level  —  in  this  form  called  the  striding- 
level  —  is  shown  resting  on  the  pivots.  The  micrometer 


FIG.  20 


THE   TRANSIT    INSTRUMENT  125 

box  can  be  rotated  90°  to  make  the  movable  wire  perpen- 
dicular to  the  rotation  axis  for  the  transit  instrument,  or 
parallel  to  it  for  the  zenith  instrument.  The  cut  shows 
the  micrometer  in  the  latter  position.  A  diagonal  eye- 
piece enables  the  instrument  to  be  used  with  zenith  stars. 
The  small  circles  attached  to  the  sides  of  the  telescope  are 
used  for.  setting  the  telescope  at  the  zenith  distance  or 
altitude  of  the  star  to  be  observed.  The  vernier-arms  bear 
both  coarse  and  delicate  levels.  When  one  of  the  verniers 
is  set  at  the  proper  reading  for  the  star,  the  telescope  is 
moved  in  altitude  until  the  bubble  of  the  coarse  level 
"plays."  The  star  will  then  pass  through  the  approxi- 
mate center  of  the  eyepiece.  It  is  made  to  pass  between 
the  two  horizontal  wires  by  turning  the  slow-motion 
screw. 

Another  common  form  of  the  transit  instrument  is  that 
in  which  one  end  of  the  axis  is  made  to  take  the  place  of 
the  lower  half  of  the  telescope.  A  prism  is  placed  at  the 
intersection  of  the  telescope  and  axis.  This  turns  the  rays 
of  light  through  90°  to  the  eyepiece,  which  is  in  one  end 
of  the  axis.  This  form  is  sometimes  called  the  broken  or 
prismatic  transit. 

An  excellent  form  of  the  prismatic  transit  is  shown  in 
Fig.  21.  In  this,  the  combined  telescope  and  rotation  axis 
is  mounted  east  and  west,  and  the  totally  reflecting  prism 
is  immediately  in  front  of  the  object  glass.  It  is  provided 
with  a  reversing  apparatus,  with  a  micrometer  and  delicate 
zenith  level,  and  can  be  used  also  as  a  zenith  telescope. 
This  instrument  is  very  compact,  and  therefore  well 
adapted  for  use  in  exploration  or  other  cases  where  trans- 
portation is  difficult. 

There  are  several  considerations  affecting  all  forms  of 
the  transit  instrument;  viz.: 

The  instrument  should  be  reversible  without  appreci- 
able jarring. 

The   lamps   used   for   illuminating   the  reticle,  or  for 


126  PRACTICAL   ASTRONOMY 

lighting  the  observing  room,  must  be  so  placed  that  they 
will  not  heat  the  instrument  appreciably. 

The  supporting  pier  must  be  isolated  from  the  floor  of 
the  observing  room,  and  should  extend  down  to  a  firm 
rock  or  soil  foundation. 

The  observing  room  should  be  constructed  so  that  it  may 
be  thoroughly  ventilated  before  observations  are  begun. 

A  sidereal  chronometer  or  clock  is  a  necessary  com- 
panion of  the  transit  instrument.  Refined  observations 


FIG.  21 

should  be  made  by  the  chronographic  method,  described 
in  §  68.  In  a  fixed  observatory,  the  clock  should  not  be 
mounted  in  the  transit  room,  but  in  an  interior  room  of 
more  constant  temperature,  where  it  can  be  placed  equally 
well  in  the  electric  circuit. 

91.  The  transit  instrument  may  be  mounted  so  that  its 
collimation  plane  is  either  in  the  prime  vertical,  or  in  the 
meridian.  In  the  first  case  it  may  be  used  to  determine 
the  latitude ;  but  this  method  is  practically  superseded 
by  that  of  the  zenith  telescope,  to  be  described  later. 
Mounted  in  the  meridian,  it  is  employed  in  connection 
with  a  sidereal  clock  or  chronometer  to  determine  the 


THE   TRANSIT   INSTRUMENT  127 

time,  the  right  ascensions  of  the  stars  or  other  celestial 
objects,  and  the  longitude  of  the  observer,  when  great 
accuracy  is  required;  and  we  shall  treat  only  this  case. 
Let  us  suppose  that  the  axis  is  mounted  due  east  and 
west  and  that  the  middle  wire  is  exactly  in  the  collimation 
plane.  If  the  image  of  a  star  whose  apparent  right  ascen- 
sion is  a  is  observed  on  the  wire  at  the  chronometer  time 
0f,  the  chronometer  correction  A0  is  given  by  (neglecting 
diurnal  aberration) 

A0  =  a  -  0'.  (214) 

The  observer  may  adjust  his  instrument  as  accurately 
as  he  pleases,  but  the  adjustments  will  not  remain,  owing 
to  changes  of  temperature,  strains,  etc.  It  is  customary 
to  put  the  instrument  very  nearly  in  the  meridian  when  it 
is  first  set  up,  and  thereafter  to  vary  the  adjustments  only  at 
long  intervals  of  time.  In  general,  therefore,  the  star  will 
be  observed  when  it  is  slightly  to  one  side  of  the  meridian. 
A  determination  of  the  errors  of  adjustment  of  his  instru- 
ment enables  the  observer  to  reduce  the  chronometer  time 
of  observation  to  the  chronometer  time  of  meridian  pas- 
sage ;  whence  the  chronometer  correction  is  given  by  (214) 
as  before. 

92.  Theoretically,  the  rotation  axis  should  be  in  the 
prime  vertical  and  in  the  horizon,  and  the  middle  wire 
should  be  in  the  collimation  plane. 

The  azimuth  constant,  a,  is  the  angle  which  the  rotation 
axis  makes  with  the  prime  vertical.  It  is  +  when  the  west 
end  of  the  axis  is  too  far  south. 

The  levef  constant,  6,  is  the  angle  which  the  rotation  axis 
makes  with  the  horizon.  It  is  +  when  the  west  end  of  the 
axis  is  too  high. 

The  collimation  constant,  c,  is  the  angle  which  a  line 
through  the  middle  wire  and  the  optical  center  of  the 
object  glass  —  called  the  line  of  sight  —  makes  with  the 


128  PRACTICAL   ASTRONOMY 

collimation  plane.  It  is  +  when  the  middle  wire  is  west 
(in  the  eyepiece)  of  the  collimation  plane. 

It  is  required  to  correct  the  time  of  observation  of  a  star 
for  the  small  deviations  a,  b  and  c. 

Let  SWINE  in  Fig.  22  represent  the  celestial  sphere 
projected  on  the  horizon,  Z  the  observer's  zenith,  N8  the 
meridian,  WE  the  prime  vertical,  WQE  the  equator,  and  P 


the  pole.  Suppose  the  rotation  axis  of  the  instrument  lies 
in  the  vertical  circle  AZB,  and  that  the  axis  produced  cuts 
the  sphere  in  A  and  B ;  that  the  great  circle  N'Z'S'  lies 
in  the  collimation  plane ;  and  that  N" Zn S",  parallel  to 
N'Z'S1,  is  described  by  the  line  through  the  middle  wire 
and  the  center  of  the  object  glass.  When  the  stars  are 
observed  on  the  middle  wire  they  are  on  the  circle 
N"Z">S",  whereas  we  desire  to  know  the  chronometer 
time  when  they  are  on  the  meridian.  Let  0  be  such  a 
star.  The  time  required  for  the  star  to  pass  from  0  to 
the  meridian  is  equal  to  the  hour  angle  of  0  measured  from 
the  meridian  toward  the  east.  Let  r  represent  it. 

If  we  let  90°  —  m  denote  the  hour  angle   and  n  the 
declination  of  A,  we  have  by  definition, 


THE   TRANSIT  INSTRUMENT  129 

ZPA  =  90°  -  wi,  ZA  =  90°  -  6, 

PZA  =  90°  +  a,  PO  =  90°  -  8, 

PA  =  90°  -  n,  4  O  =  90°  +  c, 

PZ  =  90°  -  <£,  OP.4  =  90°  -  m  +  T. 


From  the  triangle  ZPA  we  have 

sin  n  =  sin  b  sin  <£  —  cos  b  cos  <£  sin  a,  (215) 

sin  m  cos  n  =  sin  6  cos  <f>  -f  cos  6  sin  <£  sin  a  ;  .  (216) 

and  from  OP  A 

sin  c  =  —  sin  n  sin  8  +  cos  n  cos  8  sin  (T  —  m), 
or 

sin  (T  —  m)  =  tan  n  tan  8  +  sin  c  sec  n  sec  8.       (217) 

These  equations  are  true  for  any  position  of  the  instru- 
ment, and  determine  r  when  a,  b  and  c  are  known.  But 
for  the  instrument  nearly  in  the  meridian  a,  £,  <?,  m  and 
7i  are  small,  and  the  above  equations  become 

n  =  b  sin  <£  —  a  cos  <£,  (218) 

m  =  b  cos  (f>  +  a  sin  <£,  (219) 

T  =  m  +  n  tan  8  +  c  sec  8.  (220) 

(220)  is  Bessel's  formula  for  computing  the  value  of  r. 
Eliminating  m  and  n  from  the  three  equations  we  obtain 
Mayer's  formula 


cos  8  cos  o  cos  8 

in  which  the  terms  of  the  second  member  are  the  correc- 
tions, respectively,  for  errors  of  adjustment  in  azimuth, 
level  and  collimation. 

For  convenience,  let  us  put 

A  =  8inOfr-S),       B  =  cos(<ft-S),       c  =    1   ,        (222) 
cos  8  cos  8  cos  8 

and  (221)  becomes 

T=aA+bB  +  cC.  (223) 

The  effect  of  the  diurnal  aberration  is  to  throw  the  star 
east  of  its  true  position.     It  is  therefore  observed  too  late, 


130  PRACTICAL   ASTRONOMY 

and  the  time  of  observation  must  be  diminished  by  the 
quantity,  (116), 

0".31  cos  <£  sec  8  =  (K021  cos  <£  C.  (224) 

For  greater  accuracy  the  star  is  observed  over  several 
wires.  An  odd  number  of  wires  is  always  used.  They 
are  generally  placed  very  nearly  equidistant,  or  very  nearly 
symmetrical  with  respect  to  the  middle  wire.  Were  either 
of  these  arrangements  exactly  realized  the  mean  of  all  the 
times  of  transit  would  be  the  most  probable  time  of  transit 
over  the  middle  wire.  This  never  happens,  however,  and 
it  is  necessary  to  determine  the  intervals  between  the  wires. 

Let  i  denote  the  angular  distance  between  a  side  wire 
and  the  middle  wire ;  *  I  the  interval  of  time  required  by 
a  star  whose  declination  is  B  to  pass  through  this  distance. 
From  Fig.  13,  letting  the  two  positions  of  the  micrometer 
wire  represent  the  side  wire  and  the  middle  wire,  we  have 
in  the  triangle  CS'P, 

CS'  =  i,        S'P  =  90°  -  8,         OPS'  =  7; 

and  we  can  write 

sin  /  =  sin  i  sec  8  =  sin  i  C.  (225) 

If  the  star  is  not  within  10°  of  the  pole,  it  is  sufficiently 

accurate  to  use 

7  =  isecS  =  iC.  (226) 

Suppose  there  are  five  threads  in  the  reticle,  numbered 
I,  II,  III,  IV,  V,  beginning  on  the  side  next  to  the  clamp, 
and  that  the  clamp  is  west.  Let  tv  £2,  £3,  f4,  t5,  be  the 
observed  times  of  transit  of  the  star  over  the  wires,  and 
iv  i'2,  i±,  ip  the  distances  of  the  four  side  wires  from  the 
middle  wire.f  The  five  observed  transits  give  for  the  time 

*  That  is,  the  angle  subtended  at  the  optical  center  of  the  object  glass 
by  lines  drawn  to  the  side  wire  and  to  the  middle  wire.  It  is  also  meas- 
ured by  the  interval  of  time  required  for  a  star  in  the  equator  to  pass  from 
the  side  wire  to  the  middle  wire. 

t  For  clamp  west,  i'4  and  i'5  are  negative  ;  for  clamp  east,  ii  and  iz  are 
negative. 


THE   TRANSIT   INSTRUMENT  131 

of  crossing  the  middle  wire  either  ^  +  i\  (7,  t2  +  ^'2  (7, 
£3,  £4  +  '/4  (7,  or  £5  +  i5  (7,  which  would  all  be  equal  if  the 
observations  were  perfect.  Taking  their  mean,  the  most 
probable  time  of  crossing  the  middle  wire  is 


If  we  let 

fl      _  *1  ~^~  *2  "J"  ^3  +  ^4  ^"  ^5  C227) 

and 

C  =  iL+A  +  ii±i*  (228) 

0 

the  most  probable  time  of  crossing  the  middle  wire  is 

Om  +  im  C.  (229) 

6m  is  the  time  of  crossing  a  fictitious  wire  called  the 
mean  wire,  and  imC'i&  the  reduction  from  the  mean  wire  to 
the  middle  wire. 

The  above  method  holds  good  also  in  the  case  of  an 
incomplete  transit  ;  that  is,  one  in  which  the  transits  over 
some  of  the  wires  have  been  missed.  Thus,  suppose  that 
the  wires  I  and  IV  have  been  missed.  The  three  remain- 
ing transits  give  for  the  times  of  crossing  the  middle 
wire  <2  +  iz  C,  £3,  t5  +  i5  C\  and  their  mean  is 


3 


,      2  ~"     R  £»  (230) 


and  similarly  in  other  cases. 

In  accurate  determinations  of  the  time  several  stars  will 
be  observed,  and  if  the  chronometer  has  a  sensible  rate,  the 
chronometer  corrections  at  the  several  times  of  observation 
will  be  different.  To  equalize  them,  let  00  be  some  chro- 
nometer time  near  the  middle  of  the  series  of  observations, 
let  a  star  be  observed  at  the  time  6m,  and  let  the  rate  of 
the  chronometer  be  §6.  During  the  interval  6m  —  00  the 
chronomter  loses 

(231) 


132  PRACTICAL   ASTRONOMY 

If  this  quantity  fee  campute.d  for  all  the  stars  observed  and 
ap[)lied  to  the  observed  times,  the  resulting  chronometer 
corrections  furnished  by  the  several  stars  will  be  the  cor- 
rections at  the  instant  #0,  and  with  perfect  observations 
would  all  be  equal. 

Collecting  the  expressions  (223),  (224),  (229)  and  (231), 
we  have  for  the  observed  time  of  crossing  the  meridian 
when  the  clamp  is  west, 

0'  =  Om  +  aA+bB  +  cC-  0-.021  cos  <j>  C  +  im  C  +  (Om-  00)  89 ;    (232) 

and  therefore,  by  (214), 

A0  =  a  -  [6m  +  a  A  +  bB+  (c  -0-.021  cos  <£  +  im)C+(0m-00)  8(9].       (233) 

For  clamp  east  it  is  easily  seen  that  c  and  im  change  sign ; 
otherwise  the  formula  remains  the  same. 

The  formula  has  been  deduced  for  a  star  observed  at 
upper  culmination.  For  a  star  observed  at  lower  culmina- 
tion we  have  only  to  replace  8  by  180°  —  B  in  the  factors 
A,  B  and  (7,  and  they  become 

A  _  sin  (<f>  +  8)          B  _  cos  (<fr  +  8)          p  _ 1_        x034. 

cos  8  cos  8  cos  8 

The  factors  A,  B  and  C  are  readily  computed  with  four- 
place  tables.  But  when  an  instrument  is  set  up  perma- 
nently, as  in  an  observatory,  their  values  should  be 
computed  for  every  degree  of  declination,  and  tabulated. 
For  polar  distances  less  than  15°  it  is  convenient  to  have 
them  tabulated  for  every  ten  minutes  of  declination. 

DETERMINATION   OF   THE   WIRE   INTERVALS 

93.  (a)  If  the  instrument  is  provided  with  a  micrometer 
in  right  ascension,  set  the  micrometer  wire  in  succession 
on  each  of  the  fixed  wires.*  The  differences  of  the  microm- 

*  More  accurate  readings  will  be  obtained,  as  in  many  other  cases,  by 
setting  the  micrometer  wire  on  each  side  of  the  fixed  wire  and  just  in 
apparent  contact  with  it.  The  mean  of  the  readings  in  the  two  positions 
is  the  reading  for  the  coincidence  of  the  two  wires. 


THE   TRANSIT   INSTRUMENT 


133 


eter  readings  on  the  side  wires  and  the  middle  wire  give 
the  intervals  in  terms  of  one  revolution  of  the  screw,  which 
will  have  been  obtained  by  the  methods  of  §  61. 

Example.  Friday,  1891  Feb.  20.  The  transit  instrument 
of  the  Detroit  Observatory.  Four  sets  of  micrometer  read- 
ings were  made  when  the  micrometer  wire  was  in  contact 
with  each  side  of  the  fixed  wires,  to  find  the  wire  intervals 
and  im.  The  numbers  in  the  last  line  are  the  means  of  all 
the  readings  on  the  corresponding  wires.  The  value  of 
one  revolution  of  the  screw  is,  §  61,  R  =  45".042  =  35.003. 


I 

n 

III 

IV 

V 

29.966 
30.130 

27.443 
27.620 

24.891 
25.054 

22.361 
22.523 

19.797 
19.965 

29.969 
30.131 

27.445 
27.621 

24.890 
25.058 

22.357 
22.527 

19.799 
19.970 

29.968 
30.135 

27.448 
27.618 

24.895 
25.060 

22.363 
22.528 

19.802 
19.969 

29.968 
30.135 

27.445 
27.623 

24.898 
25.062 

22.356 
22.526 

19.797 
19.969 

30.050 


27.533 


24.976 


22.443 


19.883 


*!  =  (30.050  -  24.976)  R  =  +  15a.237, 
t*2  =  (27.533  -  24.976)  R  =  +  7  .679, 
i4  =  (22.443  -  24.976)  R  =  -  7  .607 
t5  =  (19.883  -  24.976)  R  =  -  15  .294 

im  =  +  08.003  for  clamp  west, 
im  =  —  0  .003  for  clamp  east. 


>  j 

:\ 


(£>)  Observe  the  transits  of  a  close  circumpolar  star 
over  the  several  wires,  and  solve  (225)  for  the  intervals  i. 
It  is  convenient  and  sufficiently  accurate  to  use  (225)  in 
the  form 


i  =  sin  I 


cos  8 
ISsinl' 


(235^ 


134 


PRACTICAL   ASTRONOMY 


Solving  as  in  the  case  of  (159),  the  resulting  values  of  i 
will  be  expressed  in  seconds  of  time. 

Example.  Monday,  1891  March  16,  X  Ursce  Minoris 
was  observed  at  lower  culmination  with  the  transit  instru- 
ment of  the  Detroit  Observatory,  clamp  east,  as  below. 
Required  the  wire  intervals.  From  the  Amer.  Ephem., 
p.  304,  3  =  88°  57'  47".3. 


Wires 

Ghronom, 

/ 

/ 

sin  / 

i 

I 

7»    2™  37' 

-  14™  3' 

-  3°  30'  45" 

8.787222n 

-  158.245 

II 

9    35 

-    7    5 

-  1  46  15 

8.489986H 

-    7.689 

III 

16   40 

IV 

23   40 

+    70 

+  1  45    0 

8.484848 

+    7.599 

V 

80   46 

+  14    6 

+  3  31  30 

8.788762 

+  15  .299 

A  number  of  stars  should  be  observed  in  this  way,  and 
the  mean  of  all  the  results  adopted  as  the  wire  intervals. 

DETERMINATION  OF  THE  LEVEL  CONSTANT 

94.  The  level  constant  b  is  generally  found  by  means 
of  a  spirit  level,  as  explained  in  §  62.  However,  the  level 
is  applied  to  the  outer  surface  of  the  cylindrical  pivots  and 
does  not  give  the  inclination  of  the  axis,  which  passes 
through  their  centers,  unless  their  radii  are  equal. 

To  determine  the  inequality  of  the  pivots  and  the 
method  of  eliminating  it,  let  A  and  B,  Fig.  23,  be  the 

L' 


FIG.  23 


centers  of  the  west  and  east  pivots,  for  clamp  west;  M and 
M'  the  vertices  of  the  Vs  in  which  the  pivots  rest;  L 


DETERMINATION    OF    THE   LEVEL    CONSTANT 


135 


and  L'  the  vertices  of  the  Vs  of  the  level ;  and  HM '  a 
horizontal  line.  Then  B A  C  —  BAD  is  the  inequality  of 
the  pivots,  which  we  shall  represent  by  p.  If  we  let 

B'  =  the  inclination  given  by  the  level  for  clamp  west, 
B"  =  the  inclination  given  by  the  level  for  clamp  east, 
b'    =  the  true  inclination  for  clamp  west, 
b"  =  the  true  inclination  for  clamp  east, 
ft    =  the  constant  angle  HM'M, 

we  can  write 

b'  =  B'  +  p  —  ft  —  p,  for  clamp  west, 
b"  =  B"  -p  =  J3  +p,  for  clamp  east; 

and  therefore 

B"  -  B' 


P  = 


(236) 
(237) 

(238) 


When  p  has  been  determined,  the  value  of  the  level 
constant  is  given  by  (236)  or  (237). 

The  value  of  p  should  be  determined  a  large  number  of 
times,  and  the  mean  of  all  the  individual  results  adopted 
as  its  final  value.  In  making  the  observations  the  telescope 
should  be  set  at  different  zenith  distances,  to  detect  any 
variations  of  the  pivots  from  a  cylindrical  form. 

Example.  1891  Feb.  17.  The  following  observation 
was  made  on  the  pivots  of  the  Detroit  Observatory  transit 
instrument.  Required  the  inequality  of  the  pivots  and 
the  inclinations  of  the  rotation  axis.  The  value  of  one 
division  of  the  striding  level  is  d  =  1".878  =  OM25.  [See 
§63.] 


Level  Direct 

Level  Reversed 

Clamp 

Zenith 
Distance 

w 

e 

w' 

e' 

W 

N.  30° 

7.2 

9.6 

15.6^ 

1.3 

E 

S.  30 

5.2 

11.7 

13.5 

3.4 

136  PRACTICAL   ASTRONOMY 

From  (166), 

b  =  B'  =  +  2.975  d  =  +  (K372,  for  clamp  west ; 
b  =  B"  =  +  0.900  d  =  +  0'.112,  for  clamp  east. 

Substituting  these  values  in  (238),  we  find 

p  -  _  (K065 ; 
and  therefore,  from  (236)  and  (237), 

V  =  +  0».372  -  0*.065  =  +  0«.307, 
b"  =  +  0-.112  +  0*.065  =  +  0-.177. 

The  mean  of  twenty-two  determinations  of  p  for  this  in- 
strument gave  p  =  -  (K066  ±  O'.OOl. 

Another  method  of  determining  the  level  constant  is 
given  in  §  97,  (d). 

95.  We  have  supposed  that  the  V's  in  which  the  pivots 
rest  and  the  V's  of  the  level  are  equal,  as  is  usually  the 
case.  If  they  are  unequal,  let 

2  v  =  the  angle  of  the  level  V, 

2  vl  =  the  angle  of  the  V  of  the  pivot  bearing; 

and  it  can  be  shown  that  the  inequality  of  the  pivots  is 
given  by 

P  =  B"-B> «5L»i (230) 

2  sin  v  +  sin  vl 

Again,  the  pivots  may  not  be  truly  cylindrical.  If 
irregularities  are  surely  found  to  exist,  the  instrument 
should  be  returned  to  the  maker  for  improvement;  or,  if 
that  is  not  practicable,  a  table  of  corrections  may  be  con- 
structed for  all  possible  positions  of  the  telescope.  It 
should  be  emphasized  that  observations  depending  upon 
faulty  pivots  are  very  unsatisfactory.  The  pivots  fur- 
nished by  the  best  modern  instrument  makers  seldom 
show  appreciable  defects. 


DETERMINATION    OF   THE   COLLIMATION   CONSTANT      137 


96.  The  forms  and  inequality  of 
the  pivots  may  be  investigated 
very  satisfactorily  by  the  Harkness 
spherometer,  shown  in  Fig.  24. 
The  method  of  applying  it  to  the 
determination  of  irregularities  in  a 
supposed  circular  section  of  a  pivot 
is  apparent.  To  determine  the  in- 
equality p  of  two  pivots,  let 

D  =  the  difference  of  the  spherometer 
readings  on  the  two  pivots, 

P   =  the  linear  pitch  of  the  screw, 

L  =  the  distance  between  the  V's  of  the 
transit  instrument  (expressed  in 
the  same  units  as  P),  and 

2  v  =  the  angle  of  the  spherometer  V's. 

Then  it  can  be  shown  that 


DP 


(240) 


FIG.  24 


DETERMINATION   OF   THE   COLLIMATION   CONSTANT 

97.  (a)  By  a  distant  terrestrial  object.  Place  the  tele- 
scope  in  a  horizontal  position  and  select  some  well-defined 
distant  point  whose  image  is  seen  near  the  middle  wire. 
With  the  micrometer  measure  the  distance  of  the  image 
from  the  middle  wire  in  the  two  positions  of  the  instru- 
ment. Call  this  distance  D'  for  clamp  west,  D"  for  clamp 
east ;  D'  and  D"  being  positive  or  negative  according  as 
the  middle  wire  is  west  or  east  (in  the  eyepiece)  of  the 
image.  The  collimation  constant  is  then  given  by 

c  =  i  (Df  -  Z>"),  for  clamp  west.  (241) 

For  clamp  east  the  sign  of  c  is  reversed. 

Example.  Saturday,  1891  April  4.  The  following 
observations  on  a  distant  object  nearly  in  the  horizon  were 
made  with  the  transit  instrument  of  the  Detroit  Observa- 


138  PRACTICAL   ASTRONOMY 

tory.     Required  the  value  of  c.     The  value  of  one  revolu- 
tion of  the  screw  is  35.003. 

Clamp  III  Micrometer  on  image  Micrometer  on  III 

W  W.  of  image  34.067  24.792 

.065  .795 

.058  .793 

Mean   34.063  '             Mean  24.793 

E              W.  of  image                          15.750  24.794 

.744  .792 

.752  .795 

Mean   15.749  Mean   24.794 

We  have 

D'  =  (34.063  -  24.793)  R  =  +  9.270  R, 

D"  =  (24.794  -  15.749)  R  =  +  9.045  R ; 
and  therefore,  from  (241), 

c  =  +  0.112  R  =  +  0-.336,  for  clamp  west. 

(5)  By  a  collimator.  This  is  an  ordinary  telescope, 
preferably  of  the  same  size  as  the  observing  telescope, 
placed  at  the  side  of  the  observing  room,  and  mounted  on 
an  isolated  pier  in  the  line  of  sight  of  the  observing  tele- 
scope when  that  is  turned  into  a  horizontal  position. 
Spider  threads  —  usually  one  vertical  and  one  horizontal  — 
are  placed  exactly  in  the  principal  focus  of  the  collimator. 
When  the  wires  are  suitably  illuminated  by  a  light  shining 
through  the  collimator  eyepiece,  the  rays  which  radiate 
from  them  emerge  from  the  collimator  object  glass  in 
parallel  lines,  just  as  if  the  threads  were  situated  at  an 
infinite  distance.  When  the  transit  telescope  is  directed 
to  the  collimator,  the  observer  will  see  in  the  focus  of  his 
instrument  the  images  of  the  spider  threads  in  the  colli- 
mator. The  vertical  image  forms  a  perfect  mark  from 
which  to  determine  the  collimation  constant,  by  the  same 
process  as  that  described  above  (a),  for  a  distant  terrestrial 
object.  While  the  threads  in  the  collimator  are  virtually 


<    >  -- 


DETERMINATION    OF    THE   COLLIMATION    CONSTANT      139 

at  an  infinite  distance,  they  are  in  reality  only  a  few  feet 
from  the  observer,  and  thus  the  atmospheric  disturbances 
which  affect  observations  on  a  distant  terrestrial  mark  are 
eliminated. 

(tf)  By  a  circumpolar  star.  Observe  the  transit  of  a 
close  circumpolar  star  over  the  first  two  or  three  wires; 
then  quickly  reverse  the  instrument  and  observe  the  tran- 
sit over  as  many  of  the  same  wires  as  possible,  being  sure 
to  determine  the  level  constant  both  before  and  after  revers- 
ing. Reduce  the  times  of  transit  in  the  two  positions  to 
the  equivalent  times  of  crossing  the  middle  wire.  Let  9l 
and  02  be  these  times,  and  let  br  and  b"  be  the  level  con- 
stants for  clamp  west  and  clamp  east.  Then  by  (233), 
for  clamp  west, 

A0  =  a  -  0!  -  aA  -  b'B  -  cC  +  0'.021  cos  <j>  C;  * 
and  for  clamp  east 

A0  =  a  -  #2  -  aA  -  b"B  +  cC  +  O.021  cos  <f>  C. 
Subtracting  and  solving  for  c  we  obtain 

c  =  |(02  -  0t)  cos  8  +  i  (b"  -  &')  cos  (<£  -  8).  (242) 

For  lower  culmination,  8  being  replaced  by  180°  —  S, 

c  =  -  \  (0,  -  OJ  cos  8  -  I  (V  -  b')  cos  (<£  +  8).  (243) 

An  example  is  given  in  §  103. 

(d)  By  the  nadir.  If  the  telescope  be  directed  verti- 
cally downward  to  a  basin  of  mercury,  and  a  piece  of  glass 
be  placed  diagonally  over  and  close  to  the  eyepiece  in 
such  a  way  that  light  from  a  lamp  at  one  side  will  be 
reflected  into  the  telescope,  the  middle  wire  and  its  image 
reflected  from  the  mercury  may  be  seen  near  together. 
Measure  with  the  micrometer  the  distance  between  the 
middle  wire  and  its  reflected  image.  Let  M.  be  this  dis- 
tance, and  consider  'it  positive  when  the  wire  is  west  (in 

*  The  correction  for  rate  will  be  small  compared  with  the  probable 
error  of  a  transit  of  a  slowly-moving  northern  star,  and  may  be  neglected. 


140  PRACTICAL    ASTRONOMY 

the  eyepiece)  of  its  image.  If  the  rotation  axis  is  hori- 
zontal we  have  M  =  2  c  ;  but  if  there  is  a  level  constant  \ 
the  distance  is  diminished  by  2  5,  so  that  M  =  2c  —  2b;  or 

c  =  i  M  +  b.  (244) 

With  well-constructed  instruments,  the  collimation  con- 
stant usually  remains  practically  unchanged  during  a  series 
of  observations.  The  level  constant,  on  the  contrary, 
sometimes  varies  rapidly.  Further,  the  spirit  level  is  not 
always  trustworthy.  Many  excellent  observers  do  not  use 
the  striding  level,  but  determine  the  level  constant  by  the 
method  of  the  nadir,  described  above.  The  collimation  con- 
stant having  been  determined  by  one  of  the  many  available 
methods  —  usually  by  the  aid  of  two  collimators  in  the 
case  of  large  instruments  —  the  value  of  the  level  constant 
is  given  by  (244),  thus  : 

b  =  c-\M.  (245) 

If  we  wish  to  determine  both  the  level  and  collimation 
constants  by  the  method  of  the  nadir,  we  measure  the  dis- 
tances of  the  middle  wire  from  its  reflected  image  in  the 
two  positions  of  the  instrument;  calling  this  distance  + 
or  —  according  as  the  middle  wire  is  west  or  east  of  its 
image.  Let 

M1  =  the  distance  for  clamp  west, 
M  "  —  the  distance  for  clamp  east, 
V     =  the  level  constant  for  clamp  west, 
b"    =  the  level  constant  for  clamp  east. 

We  have,  for  clamp  west, 


and  for  clamp  east, 
Therefore 


-  M")  4- 


DETERMINATION    OF   THE   COLLIMATION   CONSTANT  141 

From  (236)  and  (237),  V  -  b"  =  -  2p.     Therefore 

c  =     i  (M1  -  M")  -  p,  clamp  west,  (246) 

c  =  _  £  (M'  -  M")  +  P>  clamp  east,  (247  ) 

V  =  -  i  (M1  +  M")  -  jo,  clamp  west,  (248) 

1  +  M")  +  p,  clamp  east.  (249) 


Example.  1891  July  24.  The  following  nadir  observa- 
tions were  made  with  the  transit  instrument  of  the  Lick 
Observatory.  Required  the  values  of  e,  V  and  b". 

Clamp             Micrometer  on  middle  wire  Micrometer  on  image 

W                                    11.025  10.847 

.125  .852 

Mean     11.075  Mean     10.849 

E  11.020  11.164 

.135  .166 

Mean     11.077  Mean     11.165 

The  middle  wire  was  east  of  its  image  in  both  cases. 
For  this  instrument,  p  =  -  Os.021,  R  =  25.931.    We  have 

M>  =  -  (11.075  -  10.849)  R  =  -  0«.662, 
M"=-  (11.165  -  11.077)72  =  -  0  .258. 
Therefore 

c  =  -  08.101  +  08.021  =  —  08.080,  clamp  west, 

V  =  +  0  .230  +  0  .021  =  +  0  .251, 
b"  =  +  0  .230  -  0  .021  =  +  0  .209. 

(e)  By  two  collimators.  When  the  observing  telescope  is 
large,  it  is  inconvenient  and  very  undesirable  to  determine 
the  collimation  constant  by  any  method  which  involves 
reversing.  This  is  avoided  by  using  two  collimators,  one 
north  and  the  other  south  of  the  instrument,  the  object 
glasses  of  the  two  collimators  being  turned  toward  each 
other  and  toward  the  center  of  the  transit  instrument. 
The  view  of  one  collimator  from  the  other  collimator  is 
obstructed  by  the  intervening  transit  instrument  ;  but  in 
large  instruments  apertures  are  provided  on  opposite  sides 
of  the  enlarged  central  section  of  the  transit  telescope,  so 


142  PRACTICAL   ASTRONOMY 

that  when  the  telescope  is  directed  to  the  nadir,  and  the 
coverings  of  the  apertures  removed,  the  view  is  unob- 
structed. The  vertical  thread  in  one  collimator  and  the 
horizontal  thread  in  the  other  collimator  are  usually 
movable  by  micrometer  screws. 

Let  the  vertical  micrometer  thread  in  one  collimator  be 
brought  into  exact  coincidence  with  the  fixed  vertical 
thread  in  the  other.  The  lines  of  sight  of  the  two  colli- 
mators  will  then  be  exactly  parallel,  and  the  two  vertical 
threads,  viewed  by  the  transit  telescope,  will  represent 
objects  virtually  at  an  infinite  distance  and  having  azimuths 
differing  exactly  180°.  Measure  the  distance  D'  from  the 
middle  wire  of  the  transit  reticle  to  the  image  of  the  north 
collimator  thread,  and  the  distance  D"  from  the  middle 
wire  to  the  image  of  the  south  collimator  thread,  calling 
these  distances  -f  or  —  according  as  the  middle  wire  is 
west  or  east  (in  the  eyepiece)  of  the  collimator  images. 

Then  we  shall  have 

c  =  i(Z)'  +  D").  (250) 

DETERMINATION   OF   THE   AZIMUTH   CONSTANT 

98.  The  azimuth  constant  a  can  be  determined  only 
from  observations  of  stars.  Let  two  stars  (av  S^)  and 
(a2,  S2)  be  observed.  When  all  the  constants  except  a 
have  been  determined,  the  times  of  observation  of  the  two 
stars  can  be  corrected  for  all  errors  save  the  azimuth. 
Let  0l  and  02  be  the  times  so  corrected.  Then  (233) 
reduces  for  the  first  star,  to 

A0  =  ai  -Ol-aAl, 
and  for  the  second  star,  to 

A0  =  a_9  -  02  -  aA9 

Al  and  Az  being  the  values  of  A  corresponding  to  B1  and  S2. 
Combining  these  equations,  we  obtain 


-  0* 


(251) 


MERIDIAN   MARK,    OR   MIRE  143 

It  will  be  seen  that  to  determine  a  accurately,  all  the 
other  constants  of  the  instrument  must  be  well  determined, 
since  errors  in  any  one  or  more  of  them  affect  the  values  of 
01  and  02.  If  the  instrument  is  not  mounted  in  a  very 
stable  manner,  the  right  ascensions  a±  and  a2  should  differ 
as  little  as  possible.  The  value  of  a  will  be  determined 
best  when  the  denominator  Al  —  A2  is  as  large  as  possible. 
If  both  stars  are  observed  at  upper  culmination,  one  should 
be  as  far  south  as  possible  and  the  other  as  near  the  pole 
as  possible,  in  which  case  A1  and  A2  will  be  large  and 
opposite  in  sign.  This  condition  will  be  fulfilled  still 
better  by  observing  one  star  (ox,  Sj)  at  lower  culmination 
and  the  other  (a2,  S2)  at  upper  culmination,  both  as  near 
the  pole  as  possible  and  differing  nearly  12*  in  right  ascen- 
sion. In  this  case  04  must  be  replaced  by  12/l  +  «j  and  8l 
by  180°  —  Sj  in  the  various  formulae.  Stars  observed  at 
lower  culmination  are  marked  S.  P.  (sub  polo). 


MERIDIAN   MARK,  OR   MIRE 

99.  If  a  transit  instrument  is  to  be  used  for  making  long 
series  of  observations,  as  at  a  fixed  observatory,  it  is  well 
to  have  a  permanent  meridian  mark,  or  mire,  to  assist  in 
determining  the  azimuth  constant.  The  mark  consists 
usually  of  a  minute  circular  hole  in  a  metal  plate  mounted 
on  a  firm  pier  at  a  considerable  distance  to  the  north  or 
south  of  the  instrument.  An  isolated  pier  in  the  transit 
room  carries  a  lens  whose  center  is  in  the  line  joining  the 
mark  and  the  center  of  the  transit  instrument,  the  focal 
length  of  this  lens  being  equal  to  the  distance  of  the  mark 
from  the  lens.  When  the  mark  is  illuminated  by  a  lamp 
or  electric  light  [controlled  by  a  switch  in  the  transit 
room]  placed  behind  the  metal  plate,  the  rays  which  fall 
on  the  mire  lens  will  be  transmitted  as  parallel  rays  to 
the  observing  telescope,  and  the  observer  will  see  a  well- 
defined  image  of  the  mark  in  the  focus  of  his  instrument. 


144  PRACTICAL   ASTRONOMY 

The  focal  length  of  the  mire  lens  should  be  great,  in  order 
that  the  mark  may  be  at  a  considerable  distance,  thereby 
reducing  the  angular  value  of  any  possible  motion  of  the 
mark.  The  mire  lens  for  one  of  the  instruments  at  Pul- 
kowa  has  a  focal  length  of  556  feet.  One  at  the  Lick 
Observatory  has  a  focal  length  of  80  feet.  Well  mounted 
mires  have  been  found  to  be  almost  constant  in  azimuth 
for  months  at  a  time. 

The  azimuth  of  the  transit  instrument  having  been  de- 
termined from  observations  of  a  pair  of  azimuth  stars,  by 
the  methods  of  the  preceding  section,  the  azimuth  of  the 
mire  may  be  determined  by  measuring  the  angle  between 
the  mire  and  the  middle  wire  with  the  micrometer,  and 
combining  the  result  with  the  known  collimation  and  azi- 
muth constants.  The  mean  of  a  long  series  of  such  deter- 
minations may  be  adopted  as  the  azimuth  of  the  mire ;  and 
thereafter  a  measure  of  the  angle  between  the  mire  and 
middle,  wire,  combined  with  the  known  collimation  con- 
stant, will  determine  the  azimuth  constant.  Nevertheless, 
the  observation  of  star  pairs  for  azimuth  should  be  made 
as  usual,  and  the  results  thus  obtained  combined  with 
those  obtained  from  the  mire.  The  relative  weights  to  be 
assigned  to  the  results  from  the  two  methods  will  be  evi- 
dent after  a  short  experience  with  them. 

If  the>mire  is  mounted  at  a  small  angle  /above  or  below 
the  horizon  of  the  instrument,  the  measured  angle  between 
the  mire  and  meridian  should,  in  reality,  be  multiplied  by 
sec  J,  but  that  is  a  constant  factor,  and  with  most  mires 
need  not  be  taken  into  account. 

ADJUSTMENTS 

100.  To  set  up  the  instrument,  it  should  first  be  placed 
by  estimation  as  nearly  as  possible  in  the  meridian,  and  the 
following  adjustments  made  in  the  order  indicated. 

1st.  To  bring  the  wires  in  the  common  focus  of  the  eye- 
piece and  objective,  slide  the  eyepiece  in  or  out  until  the 


ADJUSTMENTS  14."; 

wires  are  perfectly  well  defined.  Then  direct  the  telescope 
to  a  very  distant  -terrestrial  object,  or  to  a  star,  and  move 
the  tube  carrying  the  wires  and  eyepiece  until  an  image  of 
the  object  seen  on  one  of  the  wires  will  remain  on  the  wire 
when  the  position  of  the  eye  is  changed.  Polaris  is  a  good 
star  for  this  purpose,  since  its  image  will  move  very  slowly. 
When  the  wires  are  placed  satisfactorily  in  the  focus  of 
the  objective,  the  tube  carrying  them  should  be  clamped 
firmly,  and  remain  unmolested  indefinitely.  Different 
observers  will  require  only  to  alter  the  distance  of  the 
eyepiece  from  the  wires  in  order  to  bring  both  star  and 
reticle  into  focus.  This  adjustment  should  be  made  when 
the  atmosphere  is  steady. 

2d.  Make  the  level  constant  very  nearly  zero,  testing  it 
by  the  method  of  §  94. 

3d.  To  make  the  wires  perpendicular  to  the  axis,  direct 
the  telescope  to  a  well-defined  mark  and  bisect  it  with  the 
middle  wire.  Adjust  the  reticle  so  that  the  object  remains 
on  the  wire  when  the  telescope  is  rotated  on  its  axis. 
The  intersection  of  the  two  wires  of  a  collimator  furnishes 
an  excellent  mark  for  this  purpose. 

4th.  Test  the  collimation  by  the  methods  of  §  97,  (a), 
(£>),  (c?)  or  (e),  and  move  the  reticle  sidewise  until  c  is 
made  very  small. 

5th.  To  set  the  finding  circle,  direct  the  telescope  to  a 
bright  star  near  the  zenith,  whose  declination  is  S.  When 
the  star  enters  the  field  of  view  move  the  telescope  so  that 
the  star  describes  a  diameter  of  the  field,  and  clamp  the 
instrument.  If  the  circle  is  designed  to  give  the  zenith 
distances,  set  it  at  the  reading 

z  =  <£  -  S. 

It  will  then  read  correctly  for  all  other  stars,  neglecting 
the  refraction. 

6th.  To  adjust  the  instrument  in  azimuth,  direct  the 
telescope  to  a  star  near  the  zenith  whose  right  ascension 


146  PRACTICAL   ASTRONOMY 

is  ar  Observe  its  transit  over  the  middle  wire,  and  let  the 
chronometer  time  of  transit  be  0V  The  approximate  chro- 
nometer correction  is 

A0!  =  ai  -  er 

Set  the  telescope  for  a  circumpolar  star  whose  right  ascen- 
sion a2  is  a  few  minutes  greater  than  av  It  culminates 
approximately  at  the  chronometer  time 

03  =  a2  —  A0r 

Rotate  the  whole  instrument  horizontally  so  that  the  star 
is  on  the  middle  wire  at  the  instant  when  the  chronometer 
indicates  the  time  #2. 

7th.   Repeat  the  2d  adjustment. 

8th.   Repeat  the  6th  adjustment. 

9th.  The  final  adjustment  in  azimuth  should  be  tested 
by  the  method  of  §  98. 


DETERMINATION   OF   TIME 

101.  When  the  chronometer  correction  is  required  to 
be  known  very  accurately,  it  is  customary  to  observe  the 
transits  of  ten  or  twelve  stars.  The  observing  list  should 
be  made  out  very  carefully,  in  advance.  Half  the  stars 
should  be  observed  with  clamp  west,  the  other  half  with 
clamp  east,  since  any  errors  in  the  adopted  values  of  im, 
p  and  c  will  be  practically  eliminated  by  reversing  the 
instrument.  To  determine  a  well,  a  pair  of  azimuth  stars 
should  be  observed  before  reversing,  and  another  pair  after 
reversing.  The  remaining  stars  on  the  list  should  be  those 
which  culminate  near  the  zenith,  or  between  the  zenith  and 
equator ;  since  the  zenith  stars  are  affected  least  by  an  error 
in  the  adopted  value  of  &,  and  the  time  of  transit  can  be 
estimated  most  accurately  for  the  rapidly  moving  equa- 
torial stars.  There  is  no  method  of  eliminating  an  error 
in  5,  and  it  must  be  very  carefully  determined.  A  good 


DETERMINATION    OF   TIME 


147 


program  to  follow,  with  small  or  medium-sized  instru- 
ments, is 

Take  the  level  readings 

Observe  half  the  stars 

Take  the  level  readings 

Reverse  the  instrument 

Take  the  level  readings 

Observe  half  the  stars 

Take  the  level  readings 

If  there  is  time  between  the  stars  for  making  further  level 
readings,  they  should  be  made.  In  reversing,  the  instru- 
ment should  be  handled  very  carefully  to  avoid  changing 
the  constants. 

102.   Example.    Wednesday,  1891  Feb.  25.    The  follow- 
ing observing  list  was  prepared  and  the  stars  observed  by 


No, 

Object 

Mag, 

a 

5 

Setting 

(1) 

Level 

(2) 

TT  Cephei,  S.  P. 

4.6 

23*    4m  20* 

74°  47'.9 

N.  62°  55' 

(3) 

8  Leonis 

2.3 

11     8    20 

21     7 

S.  21    10 

v  Ursce  Majoris 

3.3 

12    37 

33  41 

S.    8  36 

(5) 

(r  Leonis 

4.1 

15    32 

6  38 

S.  35  39 

A  Draconis 

3.3 

25     0 

69   56 

N.2f^39 

(7) 

Level 

*T>  jf/ 

Reverse 

(8) 

Level 

00 

X  Ursce  Majoris 

3.8 

40    19 

48   23 

N.    6     6 

(10) 

(3  Leonis 

2.0 

43    31 

15   11 

S.  27     6 

(11) 

P  Virginis 

3.3 

45      2 

2   23 

S.  39   54 

(12) 

y  Ursce  Majoris 

2.3 

48      8 

54   18 

N.  12     1 

(13) 

Level 

(14) 

c  Corvi 

3.0 

12    4    32 

—  22     1 

S.  64   18 

(15) 

4  H.  Draconis 

4.6 

7    12 

78   13.2 

N.  35  56 

(16) 

Level 

the  eye  and  ear  method  with  the  transit  instrument  of  the 
Detroit  Observatory,  to  determine  the  correction  to  side- 
real chronometer  Negus  no.  721.  The  stars  were  selected 
from  the  list  in  the  Berliner  Astronomisches  Jahrbuch,  pp. 


148 


PRACTICAL   ASTRONOMY 


190-327.  For  convenience  in  referring  to  them  in  the  ra 
ductions  they  are  numbered,  together  with  the  level  obser- 
vations, in  the  first  column.  Their  magnitudes  are  given 
in  the  third  column.  The  "Setting"  is  the  reading  at 
which  the  circle  is  to  be  set  for  observing  each  star.  The 
circle  of  this  instrument  reads  zero  when  the  telescope 
points  to  the  zenith  and  the  degrees  are  numbered  in  both 
directions  from  the  zero.  The  setting  is  therefore  the 
zenith  distance. 

The  level  observations  and  their  reductions  are 


(1) 

(7) 

(8) 

(13) 

(16) 

W.       E. 

W.        E. 

W.       E. 

W.       E. 

W.        E. 

15.1       9.1 

15.1       9.3 

15.3       9.0 

14.3     10.3 

14.6     10.1 

12.0     12.1 

13.7     10.8 

8.6     15.9 

11.0     13.5 

11.0     13.5 

12.1     12.1 

13.6     10.8 

9.0     15.4 

10.7     13.9 

11.1     13.5 

15.2       8.9 

14.8      9.5 

15.1       9.4 

14.0     10.5 

14.4     10.1 

B'  +  1  .525d 

+  2  .100^ 

B"  -  0  .212d 

+  0  .225rf 

+  0  .487^ 

B'  +  OM91 

+  0«.262 

B"  -  0'.026 

+  Os.028 

+  0'.061 

p    -0*.066 

-  0'.066 

p     -  0«.066 

-  08.066 

-  0".086 

b'   +  0«.125 

+  OM96 

V   +0«.040 

+  0".094 

+  0".127 

The  times  of  transit  over  the  five  wires  are  given  below. 


01, 

Object 

I 

II 

III 

IV 

V 

em 

b 

W 

0) 

s 

s 

s 

s 

h  m     s 

h   m     s 

+  OM25 

ft 

(2) 

43.9 

,14.6 

45.3 

16.0 

10  48  47.4 

10  49  45.44 

.136 

l( 

(8) 

26.8 

35.0 

43.0 

51.1 

53  59.3 

53  43.04 

.146 

It 

(4) 

41.8 

50.9 

0.0 

9.2 

58  18.3 

58    0.04 

.156 

u 

(5) 

40.0 

47.6 

55.3 

3.0 

11    1  10.8 

11    0  55.34 

.164 

it 

(6) 

37.1 

59.3 

21.5 

43.9 

11    6.1 

10  21.60 

.188 

u 

(7) 

.11)6 

E 

(8) 

.040 

n 

(9) 

5.6 

54.3 

42.9 

31.2 

25  19.6 

25  42.72 

.057 

11 

(10) 

10.2 

2.4 

54.5 

46.5 

28  38.6 

28  54.44 

.067 

M 

01) 

40.7 

33.2 

25.6 

17.9 

30  

30  29.35 

.072 

t( 

(12) 

57.2 

44.2 

31.1 

18.0 

33    4.8 

33  31.06 

.080 

It 

(13) 

.094 

it 

(14) 

12.1 

3.9 

55.6 

47.4 

49  39.3 

49  55.66 

.114 

U 

(15) 

49.2 

11.5 

34.2 

56.4 

51  19.4 

52  34.14 

.120 

II 

(16) 

.127 

DETERMINATION    OP   TIME  149 

For  clamp  east,  and  for  lower  culmination  clamp  west,  the 
transits  occurred  in  the  order  V,  IV,  III,  II,  I.  The  mean 
of  the  observed  times  is  given  in  the  column  Om.  The 
values  of  b  are  those  found  by  interpolating  for  the  instant 
of  observation,  assuming  its  value  to  vary  uniformly  with 
the  time  between  two  consecutive  determinations. 

Observations  for  determining  the  collimation  constant 
c  were  made  by  the  method  (a),  §  97,  on  the  preceding 
afternoon  and  the  following  forenoon,  which  gave  for 
clamp  west  c  =  +•  0s. 112  and  c  —  +•  Os.108,  respectively. 
We  shall  adopt  their  mean,  c  —  +-  Os.110. 

The  hourly  rate  of  the  chronometer  was  4-  0*.  15.  Let 
it  be  required  to  determine  the  chronometer  correction  at 
the  chronometer  time  0Q  =  llft  20™,  which  is  approximately 
the  mean  of  the  observation  times.  The  correction  for 
rate  is  (6m  -  11"  20m)  OM5. 

For  convenience,  let  c  —  Os.021  cos  <£  4-  im  =  c1,  and  we 

have 

c'  =  +  OM10  -  08.015  +  08.003  =  +  08.098,  for  clamp  west, 
c1  --  O'.llO  -  08.015  -  08.003  =  -  08.128,  for  clamp  east. 

Star  (11)  was  observed  over  only  the  first  four  wires. 
In  this  case  im  =  —  %  (i\  4-  i2  4-  i'4)  =  —  3s.  8 2 7,  and 
c'  =  -  OU10  -  Os.015  -  35.827  =  -  35.952. 

The  values  of  A,  B  and  0  are  taken  from  a  table  com- 
puted for  the  latitude  of  the  Detroit  Observatory.  They 
are  used  here  to  three  decimal  places ;  for  ordinary  work 
two  are  sufficient.  To  illustrate  the  application  of  (222) 
and  (234),  we  shall  compute  J.,  B  and  C  for  the  stars 
(2)  and  (3). 

(2)  (2)  (3)  (3) 

8    74°47'.9    A  +3.395  8    21°  7'     A  +0.387 

<£    42  16 .8    B  - 1.736  <f>    42  17     B  +  1.000 

sin  (<f>  +  8)  9.9496  C  -  3.813  sin  (<£  -  8)  9.5576  C  +  1.072 
cos  (<£  +  S)  9.6582n  cos  (<£-8)  9.9697 

sec  8      0.5813  sec  8    0.0302 

The  apparent  right  ascensions  are  taken  as  accurately 
as  possible  from  the  Jahrbuch.  We  are  now  prepared  to 


150 


PRACTICAL   ASTRONOMY 


fill  in  the  columns  A,  B,  (7,  Rate,  c' C,  bB  and  a;   after 
which  we  can  determine  a*  and  thence  a  A  and  6'. 


Star 

A 

B 

C 

Bate 

c'C 

bB 

aA 

e' 

a 

A0 

Wt, 

8 

8 

s 

s 

h  m    s 

h  m     s 

m       8 

(2) 

+  3.395 

-1.736 

-3.813 

-.08 

-0.37 

-.24 

-1.36 

10  49  43.39 

11    420.01 

+  1436.62 

0 

(3) 

+   .387 

+  1.000 

+  1.072 

-.07 

+   .11 

+  .15 

-  .15 

53  43.08 

8  19.66 

36.58 

2 

(4) 

+   .179 

+  1.187 

+  1.200 

-.06 

+   .12 

+  .19 

-   .07 

58    0.22 

12  36.77 

36.55 

2 

(5) 

+   .587 

+   .818 

+  1.007 

-.05 

+   .10 

+  .13 

-  .23 

11    055.29 

1531.78 

36.49 

2 

(6) 

-1.353 

+  2.582 

+  2.915 

-.02 

+   .29 

+  .49 

+   .54 

10  22.90 

24  59.52 

36.62 

1 

(9) 

-   .160 

+  1.497 

+  1.506 

.01 

-   .19 

+  .09 

+   .06 

25  42.69 

40  19.26 

36.57 

2 

(10) 

+   .472 

+   .922 

+  1.086 

+  .02 

-  .13 

+  .06 

-   .16 

28  54.23 

43  30.82 

36.59 

2 

(11) 

+   .642 

+   .768 

+  1.001 

+  .03 

-3.95 

+  .06 

-   .22 

30  25.27 

45    1.77 

36.50 

2 

(12) 

-  .357 

+  1.676 

+  1.714 

.03 

-  .22 

+  .13 

+   .12 

3331.12 

48    7.67 

36.55 

1 

(14) 

+   .972 

+   .468 

+  1.078 

+  .07 

-   .14 

+  .05 

-   .34 

49  55.30 

12    431.81 

36.51 

1 

(15) 

-2.875 

+  3.966 

+  4.898 

.08 

-  .63 

+  .48 

+  1.00 

52  35.07 

7  11.5S 

+  14  36.51 

0 

Using  stars  (2)  and  (6)  to  determine  a  we  have 


ai  =  ll*  4OT208.01, 
Ol  =  10  49  44 .75, 
01==+  14  35.26, 
A.=  +  3.395, 


a2  =  ll»24»59'.52, 
$1  =  11  10  22  .36, 
e2  =  +    14   37  .16, 
A2  =  -    1.353; 


and  therefore,  from  (251),  a  —  —  Os.400.  Similarly,  from 
(14)  and  (15)  we  obtain  a  =  —  (K348.  Using  these  as 
the  values  of  a  for  clamp  west  and  east  respectively,  we 
form  the  column  a  A.  All  the  corrections  have  now  been 
computed.  Substituting  them  in  (233)  for  each  star,  we 
obtain  the  values  A0. 

Stars  (2)  and  (15)  were  observed  solely  to  determine  a, 
and  the  values  of  A0  furnished  by  them  will  be  given  a 
weight  0,  in  the  last  column.  Assigning  a  weight  2  to  the 
stars  which  culminate  near  the  zenith  and  between  the 
zenith  and  equator,  and  a  weight  1  to  those  outside  these 
limits,  for  the  reasons  given  in  §  99,  we  obtain  for  the 
weighted  mean  of  the  chronometer  corrections, 

A0  =  +  14"  36«.55  ±  0-.009, 

which  we  shall  adopt  as  the  chronometer  correction  at  the 
time  00  =  11*  20m. 


DETERMINATION   OF    TIME  151 

103.  To  illustrate  the  determination  of  c  by  the  method 
of  §  97,  (c),  Polaris  was  observed  at  lower  culmination 
the  same  night,  as  below. 

Polaris  Level 

Clamp  W  V  12*  52™   6«  Clamp  W  Clamp  E 

«        «  IV  12  57  51  WE  WE 

Reversed  13.9     10.2  9.9     14.0 

Clamp  E  III  13  3  22  12.0    12.0  11.4     12.5 

"        "  IV  13  9  3  12.1     12.0  11.4    12.4 

«        "  V  13  14  54  13.8    10.2  9.8    14.0 

The  intervals  of  time  required  for  Polaris  to  pass  from 
V  to  III  and  from  IV  to  III  are  given  by  (225)  first  put- 
ting it  in  the  form 

sin  /  =  15  sin  1"  sec  8.  i. 

The  value  of  8  was  +  88°  43'  49".     Substituting  i'4  =  75,607 
and  i5  =  15S.294  successively  for  i  in  the  formula,  we  find 

74  =  1°  25'  50"  =  5™  43'.3,  75  =  2°  52'  37"  =  11"  30-.5  ; 

and  therefore-  the  equivalent  times  of  transit  over  III  are 

Clamp  W 

12*  52™    6*  +  Hm  30'.5  =  13*  3*  36«.5 
12  57    51  +/6    43.3  =  13  3    34.3 


Clamp  E 

13*    3"*  22*  =  13*  3"»  22«. 

13     9      3  -    5"  43'.3  =  13  3    19  .7 
13  14    54  -  11    30  .5  =  13  3    23  .5 

Taking  the  means  for  clamp  west  and  clamp  east,  we  obtain 
0!  =  13ft  3"  35».4,  02  =  13*  3"*  2K7. 

The  level  constants  given  by  the  above  observations  are 
y=  +  0-.050,  6"=-0«.096. 

Substituting  these  in  (243)  we  obtain 
c  =  +  0-.091,  clamp  west. 


152  PRACTICAL  ASTRONOMY 


REDUCTION   BY    THE  METHOD   OF   LEAST   SQUARES 

104.  In   case   the   chronometer   correction   is   required 
with  all  possible  accuracy,  the  series  of  transit  observations 
should  be  reduced  by  the  method  of  least  squares.     Let  us 
assume  that  the  level  constant,  the  rate  and  im  are  accu- 
rately determined,  and  that  the  chronometer  correction, 
the  azimuth  constant  and  the  collimation  constant  are  to 
be  obtained  from  the  observations.     To  avoid  dealing  with 
large  quantities,  let  A00  be  an  approximate  value  of  A0, 
and  x  a  small  correction  to  A00,  such  that 

A00  +  x  =  A0.  (252) 

Further,  let 

A00  +  Om  +  bB  -  (K021  cos  <£C  +  imC  +  (Om  -  00)  80  -  a  =  d.  (253) 
Then  (233)  takes  the  form 

aA  ±  cC  +  x  +  d  =  0,  (254) 

the  lower  sign  being  for  clamp  east.  A  value  for  A#0 
having  been  assumed,  all  the  terms  in  (253)  are  known 
for  each  star.  Therefore,  a,  c  and  x  are  the  only  unknown 
quantities  in  (254).  Each  star  furnishes  an  equation  of 
this  form,  and  their  solution  by  the  method  of  least 
squares  gives  the  most  probable  values  of  #,  c  and  x\ 
and  therefore,  by  (252),  the  most  probable  value  of  A#. 

105.  The  accuracy  with  which  the  time  of  transit  of  a 
star  over  a  wire  can  be  estimated  depends  upon  the  power 
of  the  instrument  and  the  declination  of  the  star.     As- 
sistant Schott  of  the  Coast   Survey*  discussed   a  large 
number  of  observations,  and  found  that  the  probable  error 
of  the  observed  time  of  transit  over  one  wire  is  best  repre- 
sented by 


c  =  V  (0.063)2  +  (0.036)2  tan2  8,  for  large  instruments, 


e  =  V  (0.080)2  +  (0.063)2  tan2  8,  for  small  instruments. 
*  See  U.  S.  Coast  and  Geodetic  Survey  Report  for  1880. 


REDUCTION  BY  THE  METHOD   OF   LEAST   SQUARES      153 

The  values  of  e  given  in  the  table  below  are  computed 
from  these  for  the  different  values  of  8.  If  1  be  the 
weight  of  an  observation  of  an  equatorial  star,  e0  its 
probable  error,  and  p  the  weight  of  an  observation  of 
any  other  star  we  have,  from  theory, 


For  large  instruments,  e0  =  0*.063,  and  for  small  ones, 
e0  =  05.080.  •  Substituting  the  values  of  €  in  this  equation 
we  find  the  following  values  of  p. 


1 

Large  Instruments 

Small  Instruments 

e 

p 

Vp 

e 

P 

V£ 

0° 

±  0-.06 

1.00 

1.00 

±  O'.OS 

1.00 

1.00 

10 

.06 

1.00 

1.00 

.08 

.98 

1.00 

20 

.06 

.98 

1.00 

.08 

.92 

.96 

30 

.07 

.91 

.95 

.09 

.83 

.91 

40 

.07 

.82 

.90 

.10 

.70 

.83 

50 

.08 

.69 

.83 

.11 

.53 

.73 

55 

.08 

.61 

.78 

.12 

.44 

.66 

60 

,09 

.51 

.71 

.14 

.34 

.59 

65 

.10 

.40 

.63 

.16 

.26 

.51 

70 

.12 

.29 

.54 

.19 

.18 

.42 

75 

.15 

.18 

.43 

.25 

.10 

.32 

80 

.21 

.09 

.30 

.37 

.05 

.22 

85 

.42 

.02 

.15 

.72 

.01 

.11 

86 

.52 

.015 

.122 

.90 

.008 

.088 

87 

.69 

.008 

.091 

1.21 

.004 

.066 

88 

1.03 

.004 

.061 

1.82 

.002 

.044 

89 

2.06 

.001 

.031 

3.70 

.000 

.022 

90 

00 

.000 

.000 

GO 

.000 

.000 

The  observation  equations  (254)  should  be  multiplied 
through  by  the  square  roots  of  their  respective  weights 
before  forming  the  normal  equations.  (254)  becomes 


Vp  (aA  ±  cC  +  x  +  d)  =  0. 


(255) 


154 


PRACTICAL  ASTRONOMY 


In  case  some  of  the  wires  have  been  missed,  the  weight 
is  diminished.     If  we  let 

N  =  the  whole  number  of  wires, 

n   —  the  number  of  wires  observed, 

1    =  the  factor  for  an  observation  over  the  JV  wires, 

P  —  the  factor  for  an  observation  over  n  wires, 

then  the  weight  for  an  incomplete  transit  is  pP. 

Assistant  Schott  found  that  we  should  use 

1.6 


P  = 


P  = 


for  large  instruments, 


,  for  small  instruments. 


(256) 


(257) 


The  following  table  gives  the  value  of  P  for  reticles  con- 
taining seven  and  five  wires,  for  the  different  values  of  n. 


Large  Instruments 

Small  Instruments 

n 

P 

n 

P 

n 

P 

n 

P 

7 

1.00 

5 

1.00 

7 

1.00 

5 

1.00 

6 

.97 

4 

.94 

6 

.96 

4 

.93 

5 

.93 

3 

.86 

5 

.92 

3 

.84 

4 

.88 

2 

.73 

4 

.86 

2 

.70 

3 

.80 

1 

.51 

3 

.77 

1 

.47 

2 

.68 

2 

.64 

1 

.47 

1 

.43 

106.  We  shall  now  apply  these  methods  to  the  reduction 
of  the  transit  observations  in  §  102. 

We  shall  assume  A00  =  +  14™  365.5.  The  values  of 
Om,  bB,  (9m  —  BQ)  80,  and  a  are  obtained  as  before,  and  we 
shall  use  their  values  tabulated  in  §  102.  To  compute  the 
terms  -  05.021  cos  <f>  0  and  imC,  let  -  05.021  cos  <f>  +  tm  =  c". 
Then 


REDUCTION   BY    THE   METHOD   OF   LEAST   SQUARES      155 


c"  =  -  O'.OIS  +  (K003  =  -  0«.012,  for  clamp  west, 
c"  -  _  o  .015  -  0  .003  =  -  0  .018,  for  clamp  east, 
c"  =  -  0  .015  -  3  .827  =  -  3  .842,  for  star  (11). 

The  products  c"  O  are  given  in  the  table  below.  The 
value  of  d  is  found  for  each  star  by  (253).  The  column 
Vp  is  taken  from  the  table  for  the  large  instruments ;  but 
for  star  (11),  which  is  incomplete,  the  square  root  of  the 
weight  is  found  from  pP. 


Star 

c"C 

d 

vt 

(2) 

+  0».05 

+  1-.66 

0.43 

(3) 

-     .01 

-     .05 

.99 

(4) 

-     .01 

-     .11 

.93 

(5) 

-     .01 

+     .13 

1.00 

(6) 

-     .03 

-     .98 

.54 

(9) 

-     .03 

+     .03 

.84 

(10) 

-     .02 

+     .18 

1.00 

(11) 

-3.84 

+     .33 

.97 

(12) 

-     .03 

H-     .02 

.79 

(14) 

-     .02 

+     .45 

.99 

(15) 

-     .09 

-     .47 

.34 

Substituting  the  values  of  A,  C,  d  and  Vp  in  (255), 
being  careful  to  change  the  sign  of  the  c  term  for  clamp 
east,  we  have  the  weighted  observation  equations 

+  1.462  a  -  1.640  c  +  0.43  x  +  0.714  =  0, 


.383 

+  1.061 

+    .99 

-    .049 

=  0, 

.166 

+  1.116 

-f    .93 

-    .102 

=  0, 

.587 

+  1.007 

+  1.00 

+    .130 

=  0, 

.731 

+  1.574 

+    .54 

-    .529 

=  0, 

.134 

-  1.267 

+    .84 

+    .025 

=  0, 

.472 

-  1.036 

+  1.00 

+    .180 

=  0, 

.623 

-    .971 

+    .97 

+    .320 

=  o, 

.282 

-  1.354 

+    .79 

+    .016 

=  0, 

.962 

-  1.067 

+    .99 

+    .445 

=  0, 

.977 

-  1.696 

+    .34 

-    .160 

=  0.. 

(258) 


156  PRACTICAL   ASTRONOMY 

The  normal  equations  formed  from  these  are 

+  5.780  a  -    2.278  c  +  2.714  x  +  2.331  =  0, ' 

-2.278    -f  18.020    -2.505    -2.794  =  0,  (259) 

+  2.714  •  -    2.505    +  7.689    +  0.918  =  0. 

Their  solution  gives 

a  =  -  0«.383,  c  =  +  OM15,  x  =  +  0«.053 ; 

and  therefore 

A0  =  A00  +  x  =  +  14«  36'.5  +  (K053  =  +  14"»  36«.553. 

The  weights  of  the  quantities  just  determined  are 
pa  =  4.71,  pc  =  16.80,  px  =  6.29. 

Substituting  the  values  of  a,  c  and  x  in  (258),  we  obtain 
the  residuals 


-  0.012,  -  .021,  +  .011,  +  .074,  -  .089,  -  .024,  -  .067,  +  .021,  +  .010,  +  .007,  +  .037. 

The  sum  of  the  squares  of  these  is  ^pvv  =  0.0147.  The 
probable  error  r±  of  an  observation  of  weight  unity  is 
given  by 

TI  =  ±  0.674  A^^-»  (260) 

where  m  is  the  number  of  observation  equations,  and  q  is 
the  number  of  unknown  quantities.  In  this  case  m  =  11 
and  q  =  3.  Therefore  rl  =  ±  Os.029. 

The  probable  errors  of  the  unknowns  are  given  by 


Therefore 

r.  =  ±  0*.013,  rc  =  ±  08.007,  rx  =  ±  O'.Oll, 

and 

a  =  -  0'.383  ±  0«.013, 

c  =  + 0'.115±0«.007, 
A^  =  +  14"  36'.553  ±  0-.011. 


PERSONAL   EQUATION  157 

CORRECTION   FOR   FLEXURE 

107.  In   the   broken    or    prismatic    transit    instrument 
(§  90),  a  correction  for  flexure  due  to  the  bending  of  the 
axis  must  be  applied.     The  effect  of   the   flexure   is  to 
change  unequally  the  positions  of  the  eyepiece  and  objec- 
tive, which  is  the  same  as  changing  the  inclination  of  the 
axis.     It  can  therefore  be  allowed  for  by  changing  the 
measured  inclination  6,  using 

b  +  f    for  clamp  west, 
b  —  f    for  clamp  east, 

/  being  the  coefficient  of  flexure,  and  the  eyepiece  being 
on  the  clamp  end  of  the  axis. 

It  requires  special  apparatus  to  determine  /  directly,  so 
that  unless  its  value  for  a  particular  instrument  has  been 
well  determined,  it  is  best  to  reduce  all  the  transit  obser- 
vations by  the  method  of  least  squares,  inserting  another 
unknown  quantity  /,  thus : 

V^  (aA  ±fB  ±  cC  +  x  +  d)  =  0.  (262) 

PERSONAL   EQUATION 

108.  It  generally  occurs  that  two  observers  differ  appre- 
ciably in  their  estimates  of  the  time  of  transit  of  a  star 
over  a  wire.     Some  observers  acquire  the  habit  of  noting 
a  transit  too  early,  while  others  note  it  too  late.     To  illus- 
trate, if  a  star  actually  transits  at  9S.5,  one  observer  may 
note  it  systematically  at  9*.7,  whereas  another  may  note 
it  systematically  at  9s. 2.     An  observer's  absolute  personal 
equation   is   the  quantity  which  must   be  applied   to  his 
observed  time  of  transit  to  produce  the  actual  time   of 
transit.     The  relative  personal  equation  of  two  observers 
is  the   quantity  which   must  be  applied  to  the  time  of 
transit  noted  by  one  observer  to  produce  the  time  noted 
by  the  other. 

The  personal  equation  arises  from  the  observers'  habits  of 


158  PRACTICAL   ASTRONOMY 

observation,  and  under  uniform  conditions  may  be  regarded 
as  sensibly  constant  for  short  periods,  of  time.  The  relative 
personal  equation  of  two  most  skilful  observers,  Bessel 
and  Struve,  was  zero  in  1814,  but  in  1821  it  had  increased 
to  05.8  and  in  1823  to  ls.O.  An  observer's  absolute  per- 
sonal equation  will  depend  very  considerably  upon  the 
circumstances  under  which  he  observes.  It  will  in  general 
be  different  for  observations  made  with  a  chronograph 
and  for  those  made  by  the  eye  and  ear  method ;  for  those 
made  with  a  clock  beating  seconds  and  with  a  chronome- 
ter beating  half -seconds ;  for  large  and  for  small  instru- 
ments ;  for  equatorial"  and  for  circumpolar  stars  ;  for  bright 
and  for  faint  stars ;  for  stars  and  for  the  moon's  edge ;  for 
different  positions  of  the  observer's  body ;  for  the  observer's 
different  degrees  of  fatigue ;  and  for  other  variable  cir- 
cumstances. 

It  is  seldom  that  an  observer's  absolute  personal  equa- 
tion exerts  an  injurious  effect  upon  results  obtained  in 
completed  form  from  his  own  observations.  But  when 
results  obtained  by  two  observers  are  to  be  compared  or 
combined,  it  is  often  essential  that  their  personal  equation 
be  eliminated. 

The  relative  personal  equation  of  two  observers  A  and  B 
may  be  determined  by  one  of  many  methods. 

(a)  Let  A  observe  the  transit  of  a  star  over  the  first  three 
or  four  threads  of  a  transit  instrument,  and  B  its  transit 
over  the  remaining  threads.  For  a  second  star  let  the 
observers  alternate,  B  observing  the  transits  over  the  first 
threads,  and  A  over  the  last  threads.  When  twenty-five 
or  more  stars  have  been  observed,  let  the  observations  of 
each  be  reduced  to  the  corresponding  times  of  transit  over 
the  middle  wire,  by  equation  (226).  The  difference  of 
times  thus  obtained  for  the  two  observers  will  be  their 
relative  personal  equation.  The  objection  to  this  method 
is  that  the  observers  are  liable  to  be  unduly  hurried  in 
exchanging  positions  at  the  eyepiece. 


DETERMINATION   OF  LONGITUDE  159 

(£)  Let  A  observe  a  star's  transit  over  all  the  threads  as 
usual.  Let  a  second  star's  transit  be  observed  by  B  as 
usual.  In  this  manner  let  the  observers  alternate  until 
each  has  observed  a  long  and  well  selected  list  of  stars  for 
determining  the  clock  correction.  Let  each  reduce  his 
observations  as  usual.  The  difference  of  their  clock  cor- 
rections will  be  their  relative  personal  equation. 

(<?)  Various  personal  equation  machines  have  been  de- 
vised for  measuring  personal  equation.  In  these  an  artifi- 
cial star  is  made  to  cross  a  field  of  view  arranged  with  a 
reticle  just  as  in  the  transit  instrument,  and  the  observer 
notes  the  times  of  transit  in  the  usual  manner.  The 
actual  times  of  transit  are  recorded  automatically  by  an 
electrical  device.  The  difference  of  the  times  determined 
in  the  two  ways  is  the  observer's  absolute  equation,  pro- 
vided the  machine  has  no  personal  equation  in  making  its 
automatic  record.  At  any  rate,  the  difference  of  the 
results  thus  obtained  for  two  observers  is  their  relative 
personal  equation. 

The  original  programs  of  observation  should  always 
l>e  arranged,  if  possible,  with  reference  to  the  direct  elimi- 
nation of  the  personal  equation.  For  example,  in  the  case 
of  longitude  determinations,  the  personal  equation  of  the 
observers  is  eliminated  by  their  exchanging  places  when 
the  program  of  observations  is  half  completed;  and 
similarly  in  other  cases. 

DETERMINATION  OP  GEOGRAPHICAL  LONGITUDE 

109.  The  accurate  determination  of  the  difference  of 
longitude  of  two  places  requires  the  accurate  determination 
of  the  time  at  each  place  and  a  method  of  comparing  these 
times.  One  of  the  following  methods  of  comparison  is 
generally  employed. 

(a)  By  Transportation  of  Chronometers.  Let  the  eastern 
place  be  E,  the  western  place  IF,  and  the  difference  of  their 


160  PRACTICAL    ASTRONOMY 

longitude,  L.  Determine  the  correction  A0e  and  the  rate 
80  of  a  chronometer  at  E,  at  the  chronometer  time  6e. 
Carry  the  chronometer  to  JF,  and  there  determine  its  cor- 
rection AOW  at  the  chronometer  time  Ow.  Then 

Ow  +  A0W  =  correct  time  at  W  at  chronometer  time  Ow ; 
Ow  +  A0e  +  BO  (Ow  —  Oe)  =  correct  time  at  E  at  chronometer  time  Ow. 

Their  difference  is 

£  =  A0e  +  SO  (Ow  -  0^  -  A0W.  (263) 

The  rate  of  the  chronometer  during  transportation  gen- 
erally differs  from  its  rate  when  at  rest.  The  change  may 
be  eliminated  largely  by  transporting  it  in  both  directions 
between  E  and  W.  The  rate  is  also  a  function  of  the  tem- 
perature and  the  lubrication  of  the  pivots.  It  has  been 
found  that  the  rate  m  at  any  temperature  •&  can  be  repre- 
sented by  the  formula 

m  =  mQ  +  k  ($  -  tf 0)2-  k'tt  (264) 

in  which  #0  is  the  temperature  of  best  compensation,  m0 
the  rate  at  that  temperature  with  t  =  0,  t  the  time  meas- 
ured from  that  instant,  k  the  temperature  coefficient  and 
kr  the  lubrication  coefficient.  By  determining  w0,  &,  #0 
and  kr  for  each  chronometer,  keeping  a  record  of  the  tem- 
perature during  transportation,  and  transporting  several 
chronometers  in  both  directions,  the  method  yields  good 
results.  It  should  never  be  employed,  however,  except 
when  the  telegraphic  method  is  impracticable. 

(6)  By  the  Electric  Telegraph.  To  illustrate  the  sim- 
plest application  of  the  method  first,  let  the  observers  at 
E  and  W  determine  their  chronometer  corrections.  Next, 
let  the  observer  at  E  tap  the  signal  key  of  the  telegraph 
line  joining  E  and  W  simultaneously  with  the  beats  of 
his  chronometer,  and  let  the  observer  at  W  note  on  his 
chronometer  the  times  of  receiving  these  signals.  In  the- 


DETERMINATION    OF    LONGITUDE  161 

same  way  let  the  observer  at  W  send  return  signals  to 
the  observer  at  E.     Let 

Oe    =  correct  time  at  E  of  sending  signal, 
Ow  =  correct  time  at  Wof  receiving  signal, 
OJ  =  correct  time  at  W  of  sending  return  signal, 
Oe'  =  correct  time  at  E  of  receiving  return  signal, 
p.    =  the  transmission  time. 
Then 

Oe  +  /*  -  L  =  Ow 

Oe'-n-L  =  BJ. 
Therefore 

L  =  *  (6.  +  ee>)  -  Kft.  +  ej\  (265) 

ft  =  i(0«-0«')-i(0.-0.')-  (266) 

There  are  several  small  errors  affecting  the  value  of  L 
obtained  by  this  method  of  comparison,  viz. : 

1st.  The  personal  equation  of  the  observers  in  sending 
and  receiving  the  signals  ; 

2d.  The  time  required  to  close  the  circuit  after  the 
finger  touches  the  key,  and  to  move  the  armature  of  the 
receiving  magnet  through  the  space  in  which  it  plays  — 
called  the  armature  time. 

3d.  The  personal  equation  of  the  observers  in  determin- 
ing the  chronometer  corrections,  and  errors  in  the  right 
ascensions  of  the  stars  employed. 

These  must  be  eliminated  as  far  as  possible  in  refined 
determinations.  This  is  best  done  by  a  modification  of 
the  above  method,  called  the  method  of  star  signals.  One 
clock  or  chronometer,  provided  with  a  break-circuit,  is 
placed  in  the  circuit  of  the  telegraph  line,  and  at  each 
station  a  chronograph  and  the  signal  key  of  a  transit 
instrument  are  placed  in  the  same  circuit.  The  same  list 
of  stars  is  observed  at  both  places,  thereby  eliminating 
errors  in  the  right  ascensions.  When  the  first  star  crosses 
the  wires  of  the  transit  instrument  at  J£,  the  observer 
makes  the  records  on  both  chronographs  by  tapping  his 


162  PRACTICAL   ASTRONOMY 

key.  When  the  same  star  reaches  the  meridian  of  W  the 
observer  there  makes  a  similar  record  on  both  chronographs, 
and  similarly  for  the  other  stars.  The  observers  must  also 
make  suitable  observations  for  determining  the  constants 
of  their  instruments  and  the  rate  of  the  clock.  Let 

Oe  =  the  clock  time  when  a  star  is  on  the  meridian  of  E,  from  the 
chronograph  at  jEJ, 

Oe'  =  the  same,  taken  from  the  chronograph  at  W, 

Ow  =  the  clock  time  when  the  same  star  is  on  the  meridian  of  TF,  taken 

from  the  chronograph  at  E, 

OJ  =  the  same,  taken  from  the  chronograph  at  W, 
e  =  the  absolute  personal  equation  of  the  observer  at  JBJ, 
w  —  the  absolute  personal  equation  of  the  observer  at  W, 
dO  =  the  correction  for  rate  in  the  interval  6W  —  Oe. 

Then 

Ow  +  dO  4-  w  -  p.  -  L  =  Oe  +  e, 

OJ  +  dO  +  w  +  p  -  L  =  Oe'  +  e. 
Therefore 

L  =  Kft.  +  *.')  -  K*.  +  0.')  +  dO  +  w-e, 

which  we  may  write 

L  =  Ll  +  w-e.  (267) 

If  now  the  observers  exchange  places  and  repeat  the 
observations  we  shall  obtain 

L  =  L2  +  e  --w,  (268) 

provided  their  relative  personal  equation  has  not  changed. 
Therefore, 

L  =  \(Ll  +  Za).  (269) 

Great  care  must  be  taken  in  arranging  the  circuits  to 
insure  that  the  electric  constants  are  the  same  at  both 
stations.  This  condition  can  be  secured  by  means  of  a 
rheostat  and  galvanometer  placed  in  the  circuit  at  each 
station.  If  there  is  any  doubt  as  to  the  equality  of  the 
constants,  any  difference  in  the  armature  times  at  the  two 
stations  may  be  eliminated  by  exchanging  the  electrical 


DETERMINATION    OF   LONGITUDE  163 

apparatus,  along  with  the  observers,  at  the  middle  of  the 
series. 

If  the  above  conditions  are  realized,  the  resulting  longi- 
tude will  be  free  from  all  errors  except  the  accidental 
errors  of  observation. 

The  method  of  star  signals  requires  the  exclusive  use  of 
the  connecting  telegraph  line  for  several  hours  on  each 
observing  night.  If  such  an  arrangement  is  impossible, 
the  observers  must  adopt  some  practicable  method.  Thus, 
if  the  telegraph  line  can  be  used  only  a  few  minutes  each 
night,  a  set  of  adopted  signals  can  be  sent  back  and  forth 
in  such  a  way  as  to  be  recorded  on  both  chronographs. 
The  time  at  the  two  stations  having  been  accurately  deter- 
mined, from  a  carefully  selected  list  of  stars,  the  results 
obtained  by  this  method  are  nearly  as  accurate  as  those 
obtained  by  star  signals. 

The  clock  or  chronometer  should  never  be  placed 
directly  in  the  circuit  joining  the  two  stations,  as  the  cur- 
rent would  generally  be  strong  enough  either  to  change 
its  rate  or  to  injure  its  mechanism.  It  should  be  placed 
in  a  local  circuit  of  its  own,  with  a  current  just  sufficient 
to  work  a  relay  connecting  it  with  the  main  circuit. 

If  so  desired,  a  clock  or  chronometer  may  be  connected 
with  the  circuit  at  each  station,  so  that  the  beats  of  both 
will  be  recorded  on  both  chronographs. 

It  is  the  custom  of  the  Coast  Survey  to  determine  longi- 
tudes from  observations  and  signals  on  ten  nights,  the 
observers  exchanging  places  at  the  middle  of  the  series. 

(c)  By  the  Heliotrope.  In  mountainous  regions  the 
telegraphic  method  of  determining  longitudes  is  usually 
unavailable.  The  difference  of  longitude  of  two  points  in 
sight  of  each  other  can  be  determined  from  heliographic 
signals.  The  necessity  for  sending  signals  in  both  direc- 
tions and  for  the  observers  exchanging  stations  will  be 
obvious.  The  equations  involved  will  be  similar  to  those 
in  method  £>. 


164  .          PRACTICAL   ASTRONOMY 

(c?)  By  Moon  Culminations.  The  right  ascension  of  the 
moon  is  tabulated  in  the  Ephemeris  for  every  hour  of 
Greenwich  mean  time,  whence  its  value  may  be  computed 
for  any  instant  of  time  at  a  place  whose  longitude  is  known. 
Conversely,  if  its  right  ascension  is  observed  at  a  given 
place,  the  Greenwich  time  corresponding  to  this  right 
ascension  can  be  taken  from  the  Ephemeris.  The  Green- 
wich time  minus  the  time  of  observation  is  the  longitude 
of  the  observer  west  of  Greenwich. 

The  right  ascension  is  best  observed  with  a  transit 
instrument  in  the  meridian.  An  observing  list,  containing 
two  azimuth  stars  and  four  or  more  stars  whose  declina- 
tions are  equal  to  that  of  the  moon  as  nearly  as  possible, 
is  arranged  so  that  the  moon  is  near  the  middle  of  the  list. 
The  transits  of  the  stare  and  the  moon's  bright  limb  are 
observed  in  the  usual  way.  From  the  star  transits,  the 
constants  of  the  instrument  and  the  chronometer  correction 
at  the  instant  of  observing  the  moon  are  obtained,  as  before. 
The  distance  of  the  moon's  bright  limb  east  of  the  meridian 
at  the  time  of  observation,  0,B,  is  given  very  nearly  by 
[the  neglected  effect  of  parallax  is  small  when  the  instru- 
ment is  nearly  in  the  meridian] 

r  =  aA  +  IB  +  c'C.  (270) 

The  values  of  A,  B  and  C  must  be  computed  for  the 
apparent  declination ;  that  is,  the  geocentric  declination 
minus  the  parallax,  given  by  (61).  / 

r  is  the  time  required  for  a  star  to  pass  through  the 
angle  r.  If  Aa  is  the  increase  of  the  moon's  right  ascen- 
sion in  one  mean  minute  (given  in  the  Ephemeris),  the 

mean  time  required  by  the  moon  to  describe  the  angle 
(\c\ 

T»  is  r  7^ ; —     The  sidereal  interval  is  therefore 

bO  —  Ao- 

60'164  Let  JIT         60'164 


60.164  -  Aa  60.164  -  Aa 


DETERMINATION    OF    LONGITUDE 


165 


The  values  of  log  M  can  be  taken  from  the  following 
table : 


A* 

logM 

Aa 

log  M 

Aa 

logM 

Aa 

logM 

1-.65 

0.0121 

1«95 

0.0143 

2«.25 

0.0166 

2«.55 

0.0188 

1.70 

.0124 

2.00 

.0147 

2.30 

.0169 

2.60 

.0192 

1.75 

.0128 

2.05 

.0151 

2.35 

.0173 

2.65 

.0196 

1.80 

.0132 

2.10 

.0154 

2.40 

.0177 

2.70 

.0199 

1.85 

.0136 

2.15 

.0158 

2.45 

.0181 

2.75 

.0203 

1.90 

.0139 

2.20 

.0162 

2.50 

.0184 

2.80 

.0207 

The  "  sidereal  time  of  semidiameter  passing  meridian  " 
is  tabulated  in  the  American  Epherneris,  pp.  385-392.  Let 
S  represent  it.  The  right  ascension  of  the  moon's  center 
when  on  the  meridian  is  equal  to  the  observer's  sidereal 
time  0,  and  is  given  by 


M  ±s, 


(271) 


the  upper  or  lower  sign  being  used  according  as  the  west 
or  east  limb  is  observed. 

Example.  The  moon's  east  limb  and  seven  stars  were 
observed  with  the  transit  instrument  of  the  Detroit 
Observatory,  Saturday,  1891  May  23,  to  determine  the 
longitude. 

The  star  transits  gave 

A0  =  +  15W34*.93  at  chronometer  time  16*  13"*, 
a  =  -  0*.360, 
b  =  +  0.674, 
c'  =  +  0.100. 

The  mean  of  the  observed  times  of  transit  of  the  moon's 
second  limb  over  the  five  wires  was 

Om  =  16*  12™  45-.60. 

The  moon's  geocentric  declination  =  —  22°  3', 

Parallax  =       0  51 , 

The  moon's  apparent  declination  =  —  22  54  . 


166  PRACTICAL   ASTRONOMY 

Therefore,  A  =  +  0.985,  B  =  +  0.455,   C=  +  1.085 ;  and 

r  =  0'.04  =  rM.     From  the  Ephemeris,  p.  388,  S  =  lm  95.90. 

Therefore 

a  =  0  =  16*  12»  45-.60  +  15"  34«.93  +  0-.04  -  I"1  9«.90  =  16*  27™  1CK67. 

From  the  Ephemeris,  p.  83,  the  right  ascension  at  Green- 
wich mean  time  18*  was  16"  27m  205.32.  The  difference, 
9S.65,  corresponds  to  a  difference  in  time  of  about  4W.  The 
average  increase  of  right  ascension  per  minute  during  this 
interval  was  25.3058.  The  exact  value  of  the  interval 
before  18"  is  9.65 -*- 2.3058  =  4*185  =  4m  HMO.  The 
Greenwich  mean  time  corresponding  to  the  observed  value 
of  a  was  therefore  \lh  55m  48S.90.  The  equivalent  sidereal 
time  was  22A  2m  Os.38,  and  the  longitude  of  the  observer 
was 

L  =  22»  2™  0'.38  -  16*  27«  10*.67  =  5*  34™  49'.71. 

Longitudes  obtained  by  this  method  can  be  regarded 
only  as  approximately  correct,  for  two  reasons  : 

1st.   An  error  in  the  observed  right  ascension  introduces 

fin 

an  error  —  times  as  great  in  the  resulting  longitude  ; 
Aa 

2d.  The  tables  of  the  moon's  motion  are  imperfect,  and 
the  tabulated  right  ascensions  may  be  slightly  in  error. 

This  would  introduce  an  error  about  — -  as  great   in  a 

Aa 

resulting  longitude.  The  above  example  should  be 
reduced  anew  when  the  corrections  to  the  moon's  right 
ascensions^  for  1891  are  published. 


CHAPTER  VIII 
THE  ZENITH   TELESCOPE 

110.  When  a  sensitive  spirit  level  at  right  angles  to  the 
rotation  axis,  and  a  micrometer  with  wire  moving  parallel 
to  the  axis  are  added  to  the  transit  instrument,  it  becomes 
a  zenith  telescope.     The   level   is   called   a  zenith   level. 
The  transit  instrument  and  the  zenith  telescope  are  fre- 
quently combined  in  this  way,  as  shown  in  Fig.  20. 

DETERMINATION   OF   GEOGRAPHICAL  LATITUDE 

111.  The  zenith  telescope  is  specially  adapted  to  deter- 
mining the  latitude  when  great  accuracy  is  required.     The 
method  employed  is  known  as  Talcott's  method.     It  con- 
sists in  measuring  the  difference  of  the  zenith  distances  of 
two  stars,  one  of  which  culminates  south  of  the  zenith  and 
the  other  north  of   the  zenith.     The  difference  of  their 
zenith  distances  should  not  exceed  half  the  diameter  of  the 
field  of  view,  to  avoid  observing  near  the  edge  of  the  field. 
The  difference  of  their  right  ascensions  should  not  exceed 
15m  or  20™,  to  avoid  any  change  in  the  constants  of  the 
instrument  between  the  two  halves  of  the  observation  ; 
nor  should  the  difference  be  less  than  2™  or  3™,  to  avoid 
undue  haste.     The  zenith  distances  should  never  exceed 
35°,  to  avoid  uncertainty  in  the  refractions. 

To  prepare  the  observing  list,  an  approximate  value  of 
the  latitude  must  be  known.  This  can  be  found  from  a 
map,  or  from  a  sextant  meridian  double  altitude  (§§  84-87). 

167 


168 


PRACTICAL   ASTRONOMY 


Letting  the  primes  refer  to   the   southern  star  and  the 
seconds  to  the  northern  star,  we  have 


Therefore 


(272) 
(273) 

(274) 


which  is  the  condition  that  the  two  stars  of  the  pair  must 
fulfill.  Thus,  in  latitude  42°  17',  and  with  an  instrument 
whose  field  of  view  is  40'  in  diameter,  we  must  have  two 
stars  such  that  S'  4-  £"  is  greater  than  83°  54'  and  less  than 
8»5°  14'.  A  pair  is  given  below  which  meets  these  require- 
ments. The  "  Setting  "  is  the  mean  of  the  zenith  distances. 
The  assumed  latitude  is  42°  17'. 


Star 

Mag, 

Apparent  a 

d 

z 

Setting 

K   Ursce  Majoris 

3.3 

8*  56™  12* 

+  47°  35' 

N.  5°  18' 

N.  5°  9' 

38  Lyncis 

4.1 

9  12     5 

37  16 

S.  5     1 

S.  5  9 

Care  must  be  taken,  in  forming  the  observing  list,  to 
employ  only  those  stars  whose  declinations  are  well  deter- 
mined. 

To  observe  the  first  star,  the  circle  to  which  the  zenith 
level  is  usually  attached  is  made  to  read  the  "  Setting,'' 
the  telescope  is  rotated  until  the  bubble  moves  to  the 
middle  of  the  tube,  and  the  micrometer  wire  is  moved  to 
the  part  of  the  eyepiece  where  it  is  known  the  star  will 
pass.  Thus,  in  the  pair  above,  it  is  known  that  the  first 
star  will  cross  9'  [  =  5°  18'  —  5°  9']  above  the  center.  When 
the  first  star  culminates,  or  within  a  few  seconds  of  culmi- 
nation, bisect  the  star  by  the  micrometer  wire,  and  read  the 
zenith  level  and  the  micrometer.  Reverse  the  instrument 
without  jarring  it,  bring  the  bubble  to  the  center  of  the 
level  again,  and  observe  the  second  star  in  the  same  way 
as  the  first.  It  is  sometimes  preferable  not  to  clamp  the 


DETERMINATION    OF    LATITUDE  169 

instrument  during  the  observations.  Care  must  be  taken 
not  to  change  the  position  of  the  level  with  respect  to  the  line 
of  sight  during  the  progress  of  an  observation;  the  angle 
between  the  two  must  be  preserved. 

Let  mQ  be  the  micrometer  reading  orf  any  point  of  the 
field  assumed  as  the  micrometer  zero;  ZQ  the  apparent 
zenith  distance  corresponding  to  w0  when  the  level  bubble 
is  at  the  center  of  the  tube  ;  wl  ,  m"  the  micrometer  read- 
ings on  the  two  stars,  the  readings  being  supposed  to 
increase  with  the  zenith  distance  ;  R  the  value  of  a  revolu- 
tion of  the  micrometer  screw;  b',  b'1  the  level  constants 
for  the  two  stars,  plus  when  the  north  end  is  high  ;  r',  r" 
the  refractions  for  the  two  stars.  Then  the  true  zenith 
distance  of  the  southern  star  is  given  by 


and  of  the  northern  star 

" 


0 


Substituting  these  in  (274)  and  solving  for  <£,  we  obtain 

<£  =  i  (£'+  8")  +  i(m'-m'')R  +  4(6'  +  b")  +  J(r'-  r").     (275) 


If  the  micrometer  readings  decrease  for  increasing  zenith 
distances  the  sign  of  the  second  term  is  minus. 

In  case  the  zero  of  the  level  scale  is  at  the  center  of  the 
tube, 

i  (V  +  b")  =*[(»'+  w")  -  (*'  + 


in  which  n?,  n1',  s',  sff  are  the  level  readings  for  the  two 
stars,  and  d  is  the  value  of  a  division  of  the  level. 

In  case  the  zero  of  the  level  scale  is  at  one  end  of  the 
tube, 

)  T  (n"+  «")]  I  (277) 


the  upper  sign  being  used  when  nf  is  greater  than  sr,  the 
lower  when  n'  is  less  than  sf. 


170 


PRACTICAL   ASTRONOMY 


The  refraction  correction  is  small,  and  can  be  computed 
differentially  by  the  formula 

in  which  (z'  —  z")  is  expressed  in  minutes  of  arc,  and  —  is 

the  rate  of  change  of  refraction  in  seconds  of  arc  per 
minute  of  change  in  zenith  distance.  Differentiating  (97), 

r  =  58"  tan  z, 

we  obtain  —  =  58"  sec2  z  sin  1',  (279) 

dz 

the  factor,  sin  1',  being  introduced  to  make  the  two  mem- 
bers homogeneous.  Therefore,  from  (278), 

$  (/  -  r"}  =  29"  sec2  z  sin  1'  (z'  -  z").  (280) 

The  values  of  J  (rf  —  r")  can  be  taken  from  the  following 
table,  for  the  mean  zenith  distance  z  of  the  stars.  Since 
the  micrometer  term  of  the  formula  -(275)  gives  the  approx- 
imate value  of  J  (z1  —  z"),  this  is  used  as  the  argument 
of  the  table,  rather  than  z'  —  z".  The  sign  of  this  correc- 
tion is  the  same  as  that  of  the  micrometer  correction. 

VALUES  OF  \  (r'  —  r"} 


z>  -  z" 

2  =  0° 

_  .    IA° 

„      90° 

iy          9f\° 

0  =  30° 

z  =  35° 

2 

0' 

".00 

".00 

".do 

".00 

".00 

".00 

1 

.02 

.02 

.02 

.02 

.02 

.02 

2 

.03 

.03 

.04 

.04 

.04 

.05 

3 

.05 

.05 

.06 

.06 

.07 

.08 

4 

.07  , 

.07 

.08 

.08 

.09 

.10. 

5 

.08 

.09 

.10 

.10. 

.11 

.13 

6 

.10 

.10 

.11 

.12 

.13 

.15 

7 

.12 

.12 

.13 

.14 

.15 

.18 

8 

.13 

.14 

.15 

.16 

.18 

.21 

9 

.15 

.16 

.17 

.18 

.20 

.23 

10 

.17 

.18 

.19 

.21 

.23 

.26 

11 

.18 

.19 

.21 

.23 

.25 

.28' 

12 

.20 

.21 

.23 

.25 

.27. 

.31 

3* 

3* 


DETERMINATION    OF   LATITUDE  171 

In  connection  with  the  subject  of  refraction,  a  word 
should  be  said  in  regard  to  securing  good  observing  con- 
ditions. The  observing  room  should  be  thrown  open  for 
thorough  ventilation  an  hour  or  more  before  the  observa- 
tions begin.  The  observing  room  should  assume,  as  nearly 
as  possible,  the  temperature  of  the  outside  air.  'The  line 
of  sight  should  not  pass  within  the  field  of  influence  of 
a  neighboring  chimney,  or  other  disturbing  factor.  Re- 
fined observations  should  not  be  attempted  when  the  star 
images  are  very  unsteady. 

If  for  any  reason  the  star  cannot  be  observed  at  the 
instant  of  culmination,  the  bisection  may  be  made  when 
the  star  is  at  some  distance  from  the  center  of  the  field, 
the  time  of  observation  being  noted.  The  polar  distance 
of  every  star  observed  in  this  way  will  be  too  small.*  A 
slight  correction,  called  the  reduction  to  the  meridian, 
must  be  applied.  Let  x  represent  it;  and  let  t  be  the 
distance  of  the  star  from  the  meridian  when  it  was  ob- 
served, in  seconds  of  time. 

In  the  right  triangle  formed  by  the  meridian,  the  star's 
declination  circle,  and  the  micrometer  wire  projected  on 
the  sphere,  we  have  the  side  90°  —  3  and  the  angle  t  at  the 
pole,  to  find  the  side  90°  —  (8  ±  x) .  We  can  write 

cot  (8  +  x)  —  cos  t  cot  8. 
Expanding  and  solving  for  tan  #, 

t         _  .  (1  —  cos  Q  sin  8  cos  8 
~    sin2  8  +  cos*  cos2  8  ' 

We  can  put  the  denominator  equal  to  unity  without  sensi- 
ble error,  since  t  is  always  small.     Therefore 

tan  x  =  ±  2  sin2  £*sin  8  cos  8  =  ±  \  sin  2  8  •  2  sin2 1 1 ; 
or, 

*  =  ±sin2S^i-';  (281) 

sin  I" 

the  lower  sign  being  used  for  stars  observed  near  lower 
culmination. 

*  That  is,  measured  from  the  pole  nearest  to  the  star. 


172 


PRACTICAL    ASTRONOMY 


The  correction  to  the  observed  latitude  will  always  loe^x. 
If  both  stars  of  the  pair  are  observed  off  the  meridian,  there 
will  be  two  such  terms  to  apply. 

The  values  of  x  are  tabulated  below  with  the  arguments 

8  and  t. 

VALUES  OF  x 


V 

«\ 

5s 

10- 

15« 

20s 

25* 

30" 

35s 

40' 

45 

50s 

55« 

60« 

t/ 
/* 

0° 

".00 

".00 

".00 

".00 

".00 

".00 

".00 

".00 

".00 

".00 

".00 

".00 

90° 

0 

.00 

.00 

.01 

.02 

.03 

.04 

.06 

.08 

.10 

.12 

.14 

.17 

85 

10 

.00 

.01 

.02 

.04 

.06 

.08 

.11 

.15 

.19 

.23 

.28 

.34 

80 

15 

.00 

.01 

.03 

.05 

.09 

.12 

.17 

.22 

.28 

.34 

.41 

.49 

75 

20 

.00 

.02 

.04 

.07 

.11 

.16 

.22 

.-28 

.36 

.44 

.53 

.63 

70 

25 

.01 

.02 

.05 

.08 

.13 

.19 

.26 

.34 

.42 

.52 

.63 

.75 

65 

30 

.01 

.02 

.05 

.09 

.15 

.21 

.29 

.38 

.48 

.59 

.71 

.85 

60 

35 

.01 

.03 

.06 

.10 

.16 

.23 

.31 

.41 

.52 

.64 

.77 

.92 

55 

40 

.01 

.03 

.06 

.11 

.17 

.24 

.33 

.43 

.54 

.67 

.81 

.97 

50 

45 

.01 

.03 

.06 

.11 

.17 

.25 

.33 

.44 

.55 

.68 

.82 

.98 

45 

112.  The  adjustments  for  the  transit  instrument,  §  100, 
apply  equally  well,  for  the  most  part,  to  the  zenith  tele- 
scope.    Special  forms  of  the  instrument,  however,  will  call 
for  special  methods,  which  the  intelligent  observer  will 
easily  devise. 

The  micrometer  wire  must  be  made  perpendicular  to  the 
meridian.  If  this  adjustment  is  perfect,  an  equatorial  star 
will  travel  on  the  wire  throughout  its  entire  length. 

113.  Example.    The  following  observations  were  made 
with  the  zenith  telescope  of  the  Detroit  Observatory,  Mon- 
day, 1891  March  16. 


Star 

Chronometer 

Micrometer 

Level 

n 

s 

K  Ursce  Majoris 
38  Lyncis 

8   56    10 

13.647 
37.359 

8.9 
39.6 

35.7 
12.4 

Required  the  latitude. 


DETERMINATION   OF    LATITUDE  173 

The  chronometer  correction  was  -f-  15m  47s.  The  value 
of  one  revolution  of  the  micrometer  screw  is  R^=  45".  042. 
The  value  of  one  division  of  the  level  is  d  —  2".  74.  The 
mean  places  of  the  stars  are  given  in  the  Jahrbuch,  p.  180. 
Their  apparent  places  are  found  by  the  methods  of  §  55 

to  be 

a"  =  8»  56™  12s,        8"  =  +  47°  35'  21".33, 

a'  =  9   12      5,         8'  =  +  37   15  53  .05. 
Therefore 

i  (8'  +  8")  =  42°  25'  37".19. 

The  micrometer  readings  decreased  with  increasing 
zenith  distances.  Therefore 

-  £  (ro'  -  m")  R  =  -  11.856  R  =  -  8'  54".02. 
The  zero  of  the  level  was  at  one  end  ;  therefore,  by  (277), 
£  (b>  +  b")  =  \  (52.0  -  44.6)  rf  =  +  5".07. 

The  half  difference  of  the  zenith  distances  is  8'  54",  and 
the  mean  zenith  distance  is  z  =  5°  9'.  Therefore,  from  the 
table  for  differential  refraction, 

i  (r,  _  ri,)  =  _  QH  16> 

The  first  star  was  observed  at  an  hour  angle  t  —  -f  11s; 
therefore,  from  the  table,  the  value  of  J  x  for  the  northern 

star  is 

=  +  0".02. 


The  second  star  was  observed  at  the  hour  angle  t  =  —  8s; 
therefore  the  value  of     x  for  the  southern  star  is 


Combining  the  terms  of  (275)  and  the  reductions  to  the 
meridian,  we  obtain 

$  =  42°  16'  48".ll. 

[The  known  value  of  the  latitude  is  about  42°  16'  47".3.] 

114.    In  very  accurate  determinations  of  the  latitude,  a 
number  of  pairs  of  stars  should  be  observed  several  times 


174  PRACTICAL   ASTKONOMY 

in  this  way,  and  the  results  combined  by  the  method  of 
least  squares.  If  we  let  <£0,  R0  and  c?0  be  very  nearly  the 
true  values  of  <£,  R  and  d,  and  let  A<£,  AR  and  Ac?  be 
slight  corrections  to  <£0,  RQ  and  c?0,  each  observation  fur- 
nishes an  equation  of  the  form 

<£0  +  A<£  =  I  (8' 
+  i  [±(n'  + 

Let 


+  s»Xd0-l(r'-r»)-$x  -±xn.  (282) 

Then 

A</>  -  i(m'  -  m")  A#  -  1  [±(n;  +  *OT(n"  +  5")]  Arf  +  k  =  0,    (283) 

is  an  observation  equation  for  determining  A<£,  AS  and  Ac?. 
Thus,  in  the  example  above,  if  we  assume  </>0  =  42°  16'  47".0, 
RQ  =  45".040,  d0  =  2".70,  we  find  A;  =  -  1".06,  and  (283) 

becomes 

A<£  +  11.856  A72  -  1.85  Ad  -  1.06  =  0.  (284) 

Forming  the  corresponding  equations  for  the  other  pairs 
observed  and  solving  by  the  method  of  least  squares,  the 
most  probable  values  of  A$,  A72  and  Ac?,  and  therefore 
of  <£,  R  and  c?,  are  obtained. 

However,  it  has  recently  been  shown  that  the  latitude 
of  a  place  varies  appreciably,  sometimes  in  the  course  of  a 
few  weeks  ;  and  latitude  observations,  to  be  combined  by 
any  direct  method,  must  be  made  inside  of  a  few  days  and 
the  result  be  taken  as  the  latitude  at  the  mean  of  the 
observation  times. 


CHAPTER  IX 

THE  MERIDIAN  CIRCLE 

• 
115.   The  Meridian  Circle  consists  essentially  of  a  transit 

instrument  with  a  graduated  circle  attached  at  right  angles 
to,  and  concentric  with,  the  rotation  axis.  The*  graduated 
circle  rotates  in  common  with  the  telescope,  and  is  read  by 
reading  microscopes  firmly  attached  to  one  of  the  support- 
ing piers  of  the  instrument.  The  best  instruments  are 
provided  with  two  graduated  circles.  One  of  these  circles 
usually  remains  fixed  on  the  axis  for  an  indefinite  time; 
whereas  the  other  is  movable,  and  many  observers  are 
accustomed  to  rotate  it  through  any  desired  angle  [and 
clamp  it  firmly  to  the  axis]  from  time  to  time. 

An  excellent  form  of  the  meridian  circle  is  illustrated, 
with  many  details  omitted,  in  Fig.  25.  Two  massive 
supporting  piers  extend  down  to  solid  earth  or  rock  foun- 
dation ;  and,  as  in  the  case  of  all  telescope  piers,  are  com- 
pletely isolated  from  the  floor  and  building.  The  circular 
drum  on  each  pier  carries  the  four  long  slender  reading 
microscopes  for  reading  the  graduated  circle,  on  which 
they  are  focused.  The  pivots  rest  in  Vs  attached  to  the 
inner  head  plates  of  the  drums.  The  counterbalance  levers 
are  shown  on  the  tops  of  the  drums.  A  lifting  arm  de- 
scends from  the  inner  end  of  each  lever  and  rests,  with 
roller  bearings,  on  the  under  side  of  the  axis.  The  chain 
descending  from  the  outer  end  of  the  lever,  through  a  hole 
in  the  pier,  carries  a  counterweight.  Nearly  the  whole 
weight  of  the  instrument  is  supported  in  this  manner,  leav- 
ing only  a  small  residual  weight  to  be  borne  by  the  Vs. 

175 


FIG.  25 


THE     MERIDIAN    CIRCLE  177 

The  level  is  in. position  on  the  instrument,  suspended  from 
the  pivots.  It  is  visible  immediately  under  the  circles. 
The  eyepiece  is  supplied  with  the  usual  number  of  verti- 
cal threads,  and  with  vertical  and  horizontal  micrometer 
wires.  A  basin  of  mercury,  not  visible  in  the  cut,  is 
mounted  below  the  level  of  the  floor,  immediately  under 
the  center  of  the  instrument,  on  a  pier  isolated  from  the 
floor.  The  small  telescope  showing  just  below  the  ob- 
jective of  the  lower  left  reading  microscope  is  the  "setting 
telescope,"  used  for  setting  the  instrument  at  any  desired 
circle  reading.  The  auxiliary  apparatus  shown  is,  the 
observing  chair  in  the  foreground,  the  adjustable  mercury 
basin  for  reflection  observations,  and  the  reversing  carriage 
in  the  background.  The  field  of  view  and  the  wires  are 
illuminated  by  light  from  a  lamp  at  the  side  of  the  room, 
shining  through  the  hollow  axis,  as  in  the  case  of  the 
transit  instrument.  A  system  of  small  mirrors  receives 
light  from  the  same  source  and  reflects  it  where  it  is  needed 
for  reading  -the  circles  and  setting  the  telescope.  The 
graduated  circles  are  about  two  feet  in  diameter,  and  the 
graduations  are  two  minutes  of  arc  apart.  The  micrometer 
head  of  each  microscope  is  divided  into  60  parts,  each 
division  corresponding  to  1".  The  divisions  may  be  sub- 
divided, by  estimation,  into  10  parts,  each  part  being  O'M. 
The  meridian  circle  is  used  principally  to  determine  the 
accurate  positions  of  the  heavenly  bodies  on  the  celestial 
sphere;  i.e.,  their  right  ascensions  and  declinations.  It  is 
further  adapted  to  determining  the  time  and  the  geo- 
graphical longitude  by  the  methods  of  Chapter  VII,  and 
to  determining  the  geographical  latitude. 

116.  The  determination  of  right  ascensions.  The  princi- 
ples involved  in  this  problem  have  been  treated  in  Chap- 
ter VII.  The  stars  whose  right  ascensions  are  to  be 
determined  (which  we  shall  call  undetermined  stars), 
are  placed  in  an  observing  program  which  includes  a 


178  PRACTICAL   ASTRONOMY 

considerable  number  of  stars  whose  positions  are  accurately 
known  (which  we  shall  call  standard  stars),  and  which  are 
suitable  for  determining  the  azimuth  of  the  instrument, 
and  the  time.  The  transits  of  all  the  stars,  both  unde- 
termined and  standard,  are  observed,  the  constants  of  the 
instrument  are  determined,  and  all  the  observations  are 
reduced  in  the  usual  manner.  The  clock  correction  is 
determined  from  the  standard  stars,  as  usual.  The  right 
ascensions  of  the  undetermined  stars  are  found  by  means 
of  equation  (233),  which  may  be  written 

a  =  Om  +  A0  +  aA  +  bB  +  c'C  +  (Om  -  00)  80.  (285) 

In  forming  the  observing  program,  the  undetermined 
stars  should  be  preceded,  accompanied  and  followed  by 
standard  stars.  Likewise,  the  declinations  of  the  standard 
stars  should  be  nearly  the  same  as,  or  at  least  should  in- 
clude, the  declinations  of  the  undetermined  stars,  thereby 
eliminating  largely  the  uncertainties  or  progressive  changes 
in  the  instrumental  constants. 

Example.  In  the  example  of  §  102,  let  it  be  assumed  that 
the  star  (3)  B  Leonis  and  star  (10)  ft  Leonis  are  undeter- 
mined stars,  and  that  the  remaining  nine  stars  of  the  list 
are  standard  stars.  Required  the  right  ascensions  of  stars 
(3)  and  (10). 

The  value  of  the  chronometer  correction  from  the  nine 

standard  stars  is 

A0  =  +  Um  36«.54. 

The  right  ascension  computations  may  now  be  tabulated, 

as  below. 

Star  (3)  8  Leonis  Star  (10)  ft  Leonis 

Om                     10»53~43-.04  11»28»54'.44 

A0                 -f       14  36.54  +       14   36.54 

aA                                 0 .15  0 .16 

IB                 +               0.15  +               0.06 

c'C                +               0.11  0.13 

(0m  -  00)  §0  - 0-07  + 0.02 

a                       11     8  19.62  11  43  30.77 


THE   MERIDIAN   CIRCLE  179 

117.  Declinations  and  the  latitude  are  determined  from 
observations  which  involve  readings  of  the  graduated  circle. 
The  method  of  reading  the  circle  by  reading  microscopes, 
and  correcting  for  error  of   runs,  is  given  in   §  58.     In 
modern  instruments,  provided  with  eyepiece  micrometers 
moving  in  declination,  the  error  of  runs  may  and  should 
be  practically  eliminated  by  a  suitable  method  of  observing. 
To  illustrate,  if  it  is  the  observer's  custom  to  read  each 
microscope  on  only  one  graduation,  the  telescope  should 
be  directed,  by  means  of  the  setting  telescope,  so  that  the 
graduation  to  be  set  on  will  always  fall  in  the  same  posi- 
tion with  reference  to  the  zero  of  the  microscope  for  all 
the  observations  of  a  series.     Again,  if  it  is  the  observer's 
custom  to  read  each  microscope  on  two  adjacent  gradua- 
tions, these  should  always  fall  in  the  same  positions  with 
reference  to  the    micrometer   zero,  the   two   graduations 
being,  preferably,  on  opposite  sides  of  the  zero.     Knowing 
the  approximate  position  of  the  star  to  be  observed,  its 
"setting"  —  usually  zenith  distance  —  can  be  computed  to 
the  nearest  even  minute,  and  the  instrument  set  for  that 
reading.     As  the  star  crosses  the  middle  thread  in  the  eye- 
piece, its  distance  from  the  zero  position  of  the  micrometer 
wire  is  measured  with  the  declination  micrometer.     The 
microscope  readings  on  the  graduations  are  then  secured, 
and  later  the  declination  micrometer  is  read. 

In  reading  the  circle  it  is  customary  to  take  the  degrees 
and  even  minutes  from  the  circle  as  seen  in  one  of  the 
microscopes,  and  the  seconds  and  fractions  from  the  mean 
of  the  four  microscope  readings.  The  reading  thus 
obtained  must  be  corrected  for  runs,  for  flexure,  for  the 
distance  of  the  declination  micrometer  wire  from  its  zero 
position,  and  possibly  for  errors  of  the  graduations. 

118.  The  zero  reading  of  the  micrometer  may  be  obtained 
from  nadir  observations   [§  97,  (d)].     Let  ^he  observing 
telescope  be  directed  vertically  downward  to  the  mercurial 


180  PRACTICAL  ASTRONOMY 

basin.  Obtain  the  micrometer  readings  when  the  wire  is 
on  each  side  of  its  reflected  image,  at  minute  and  equal 
distances.  The  mean  of  the  two  readings  is  the  reading 
for  coincidence  of  the  wire  and  its  image,  and  is  the  zero 
reading  of  the  micrometer.  Let  the  microscopes  be  read  for 
this  position  of  the  instrument,  and  corrected  for  runs  and 
graduation  error.  The  result  is  the  nadir  reading  of  the 
circle.  The  nadir  reading  plus  180°  is  the  zenith  reading. 
Many  of  the  older  forms  of  meridian  circles  are  not  pro- 
vided with  declination  micrometers,  but  have  two  hori- 
zontal fixed  wires  marking  the  center  of  the  field,  as  shown 
in  Fig.  19.  In  this  case,  when  a  star  is  crossing  the  middle 
transit  thread,  the  entire  instrument  is  moved  by  a  slow 
motion  screw  until  the  star  travels  midway  between  the 
two  horizontal  wires.  The  microscope  readings  may  thus 
have  any  value  up  to  2',  and  the  correction  for  runs  must 
be  carefully  determined.  Again,  some  instruments  have 
a  single  horizontal  fixed  wire. 

119.  Determination  of  the  value  of  one  revolution  of  the 
micrometer  screw.  The  method  of  §  61>  (V),  is  applicable, 
but  a  better  method  is  the  following :  Direct  the  observ- 
ing telescope  to  one  of  the  collimators  (described  in  §  97), 
so  that  the  image  of  the  horizontal  wire  of  the  collimator 
falls  about  half  a  radius  above  the  center  of  the  field. 
Determine  the  micrometer  reading  when  the  micrometer 
wire  is  coincident  with  the  image  of  the  collimator  wire, 
and  read  the  circle.  Rotate  the  instrument  so  that  the 
collimator  image  moves  to  the  opposite  side  of  the  field 
of  view,  and  again  determine  the  corresponding  micrometer 
and  circle  readings.  The  difference  of  the  circle  readings 
divided  by  the  difference  of  the  micrometer  readings  is 
the  value  of  a  revolution  of  the  screw.  If  a  movable 
circle  is  available,  several  different  arcs  may  be  used  for 
this  purpose,  thereby  eliminating  very  largely  the  effect  of 
graduation  errors. 


THE   MERIDIAN    CIRCLE  181 

120.  Eccentricity  of  the  circle.     As  explained  in  §  59, 
the  effect  of  eccentric  mounting  of  the  circle  is  eliminated 
by  the  use  of  two  or  more  equidistant  verniers  or  reading 
microscopes. 

121.  Flexure.    When  the  instrument  is  rotated  from  one 
position  to  another,  the  form  of  the  observing  telescope 
(and  s6metimes  also  the  form  of  the  graduated  circle),  is 
appreciably  changed  under  the  action  of   gravity.      The 
bending  of  the  telescope  tube  will  do  no  harm  provided 
the  objective   and  the   eyepiece   are    displaced   the  same 
amount,  but  a  difference  in  their  displacements  changes 
the  direction  of  the  line  of  sight  with  reference  to  the 
circle  graduations.     This  effect  is  called  the  flexure. 

In  most  modern  instruments  the  flexure  is  very  small, 
since  the  observing  telescope  is  symmetrical  with  refer- 
ence to  the  rotation  axis,  and  the  mechanism  at  the  eye 
end  is  made  of  the  same  weight  as  the  objective  and  its 
cell.  They  are  further  designed  so  that  the  objective  and 
eyepiece  mechanism  may  be  interchanged  on  the  telescope 
tube.  If  we  combine  two  observations  of  the  same  body, 
one  made  before  interchanging  the  objective  and  ocular, 
and  the  other  after  interchanging  them  (the  interchange 
involving  at  the  same  time  a  rotation  of  the  telescope  tube 
through  180°),  the  result  will  be  free  from  flexure,  theo- 
retically at  least. 

The  two  collimators  furnish  a  simple  method  of  measur- 
ing the  horizontal  flexure,  i.e.,  the  flexure  when  the  tele- 
scope is  in  a  horizontal  position.  Let  the  horizontal  threads 
of  the  collimators  be  brought  into  coincidence,  as  explained 
in  §  97,  (e).  Tbe  two  threads  then  represent  two  infi- 
nitely distant  lines  whose  angular  distance  apart,  measured 
through  the  zenith,  is  exactly  180°.  Measure  this  distance 
in  the  usual  manner.  If  there  is  no  flexure,  the  difference 
of  the  circle  readings  should  be  exactly  180°.  If  any 
excess  or  deficiency  exists,  that  excess  or  deficiency  is 


182  PRACTICAL   ASTRONOMY 

twice  the  horizontal  flexure,  plus  the  accidental  and  un- 
avoidable errors  of  the  observation. 

Example.  Repsold  Meridian  Circle,  Lick  Observatory, 
Saturday,  1898  June  11,  the  following  observations  were 
made  by  R.  H.  Tucker  for  determining  the  horizontal 
flexure.  Circle  east. 

Circle  Reading  on  South  Collimator     ....  224°  56'  49".07 

North  Collimator  set  on  South  Collimator 

Circle  Reading  on  North  Collimator     ....  44    56  48  .64 

North  Collimator  set  on  South  Collimator 

Circle  Reading  on  North  Collimator     ....  44    56  48  .58 

Circle  Reading  on  South  Collimator     ....  224    56  48  .75 

Mean  Circle  Reading,  North 44    56  48  .61 

Mean  Circle  Reading,  South 224    56  48  .91 

Difference,  North-South 179    59  59  .70 

The  deficiency  is  0".30  and  the  horizontal  flexure,  /,  is 
0".15.  The  sign  of  the  flexure  correction  to  the  circle 
readings  is  readily  found.  As  the  telescope  was  turned 
from  the  south  collimator  through  the  zenith  to  the  north 
collimator,  the  readings  increased  from  224°  through  360° 
to  44°,  and  the  measure  of  the  angle  is  0".30  too  small. 
The  correction  to  the  circle  reading  for  a  star  south  of  the 
zenith  is  minus,  and  for  a  star  north  of  the  zenith  is  plus. 
If  the  instrument  were  reversed,  circle  west,  the  signs  of 
the  corrections  would  be  reversed. 

The  mean  value  of  /,  resulting  from  23  determinations 
by  the  same  observer,  extending  through  two  years,  is 
0".065,  but  the  value  O'M  has  been  adopted,  provisionally. 

Since  the  gravitational  moment  of  any  given  mass  in 
the  telescope,  with  reference  to  the  rotation  axis,  varies 
with  the  sine  of  the  zenith  distance  of  the  line  of  sight,  the 
general  expression  for  the  flexure  is  assumed  to  be 

Flexure  =  /sin  z,  (286) 

though  it  is  not  probable  that  the  flexures  in  all  instru- 
ments can  be  represented  by  this  law. 


THE   MERIDIAN   CIRCLE  183 

The  order  of  observation  followed  in  the  above  example 
illustrates  a  general  principle  which  should  be  taken  into 
account,  whenever  possible,  in  forming  programs  of  ob- 
servation with  any  instrument.  The  observations  were 
made  in  one  order,  and  repeated  in  reverse  order,  thereby 
eliminating  largely  any  possible  progressive  changes  in  the 
apparatus. 

122.  Errors  of  graduation  exist  in  all  circles  and  affect 
the  angles  measured  by  them.  Whether  these  errors  are 
negligible,  or  must  be  taken  into  account,  depends  largely 
upon  the  degree  of  refinement  exacted  by  the  problem  in 
hand.  In  the  case  of  small  instruments  constructed  by 
first-class  makers,  the  errors  of  graduation  will  generally 
be  smaller  than  the  least  reading  of  the  instrument,  and 
may  be  neglected.  The  circles  provided  by  the  best  makers 
for  modern  meridian  instruments  are  nearly  perfect.  It  is 
seldom  that  one  of  the  graduations  is  displaced  as  much  as 
V  from  its  theoretical  position,  or  that  the  mean  of  four 
graduations  90°  apart  is  in  error  by  as  much  as  0".5. 
Nevertheless,  it  is  necessary  to  investigate  every  such 
circle  to  determine  the  degree  of  refinement  which  it  will 
impart  to  observations  depending  upon  its  readings,  and  to 
secure  data  for  eliminating  errors  arising  from  its  imper- 
fect graduation.  The  investigation  of  10,800  graduations 
on  a  circle  taxes  the  resources  of  most  long  established 
observatories  so  prohibitively  that  it  is  seldom  or  never 
carried  to  a  finish.  After  the  investigation  has  extended 
to  all  the  graduations  marking  the  degrees,  or  at  the 
most  to  those  marking  the  20'  divisions,  the  nature  and 
magnitude  of  the  systematic  errors  and  the  magnitude  of 
the  accidental  errors  will  have  been  revealed,  and  further 
determinations  may  generally  be  confined  to  the  gradua- 
tions which  are  used  with  special  frequency,  e.g.,  the 
graduations  used  in  determining  the  nadir  reading,  or 
those  used  with  particular  stars,  or  zones  of  stars. 


184  PRACTICAL   ASTRONOMY 

It  will  be  seen  from  the  above  that  the  problem  is  one 
for  the  professional  astronomer  and  his  assistants,  rather 
than  for  the  student.  A  complete  solution  is  therefore 
not  called  for  in  this  place,  but  an  outline  of  one  of  the 
best  methods  may  be  given. 

Let  us  suppose  that  the  instrument  has  a  fixed  and  a 
movable  circle,  each  read  by  four  microscopes  90°  apart. 
As  an  origin  for  the  entire  system  of  measures,  let  it  be 
assumed  that  the  mean  of  the  readings  of  the  four  micro- 
scopes on  the  0°,  90°,  180°  and  270°  lines  is  free  from 
graduation  error,  in  both  circles.  If  now  the  axis  of  the 
instrument  be  rotated  through  a  given  angle,  30°  for 
example,  and  the  circle  reading  be  taken,  the  observed 
angle  will  differ  slightly  from  30°,  from  several  causes : 
first,  the  unavoidable  errors  of  observation,  which  may  be 
reduced  materially  by  increasing  the  number  of  indepen- 
dent observations  ;  second,  progressive  changes  in  the  ap- 
paratus, largely  due  to  temperature  variations,  which  may 
be  reduced  materially  by  repeating  the  observations  back- 
wards ;  third,  differential  flexure  of  the  circle,  which  may 
be  eliminated,  it  is  assumed,  by  rotating  the  instrument  on 
its  axis  through  180°  and  repeating  the  observations  on 
the  same  lines ;  and  fourth,  the  graduation  errors  of  the 
divisions  used.  These  considerations  suggest  the  principal 
features  of  the  program  of  observations. 

Let  it  be  required,  first,  to  determine  the  division  errors 
of  the  45°  points  of  both  circles;  i.e.,  the  error  for  each 
circle  affecting  the  mean  of  the  readings  obtained  from 
the  four  microscopes  on  the  points  45°,  135°,  225°  and 
315°.  Place  the  0°  of  the  two  circles  in  coincidence  and 
read  the  microscopes  for  both  circles.  To  increase  the 
accuracy  of  the  determination  by  increasing  the  number 
of  observations,  and  at  the  same  time  eliminate  the  circle 
flexure,  let  these  observations  be  repeated  with  the  instru- 
ment rotated  through  one,  two,  three  and  four  quadrants. 
Now  let  the  45°  line  of  the  movable  circle  be  made  coinci- 


THE   MERIDIAN    CIRCLE  185 

dent  with  the  0°  line  of  the  fixed  circle,  and  a  series  of 
readings  similar  to  the  above  be  secured.  Next  let  the  90° 
division  of  the  movable  circle  be  made  coincident  with  the 
0°  line  of  the  fixed  circle,  and  so  on  until  a  series  has  been 
secured  with  each  45°  division  of  the  movable  circle  in 
coincidence  with  the  0°  of  the  fixed  circle.  In  order  to 
eliminate  progressive  changes  in  the  instrument,  as  far  as 
possible,  let  the  above  program  of  observations  be  re- 
peated in  reverse  order.  The  data  will  then  be  at  hand 
for  a  thorough  determination  of  the  errors  of  the  45°  divi- 
sions of  both  circles.  Each  arc  of  45°  on  one  circle  has 
been  compared  with  each  such  arc  on  the  other  circle. 
For  example,  the  first  45°  arc  of  the  fixed  circle  has  been 
used  to  measure  each  of  the  eight  45°  arcs  on  the  movable 
circle.  The  true  sum  of  these  eight  arcs  is  360°.  If  their 
sum  measured  by  the  fixed-circle  arc  differs  from  360°  by 
any  quantity  n,  the  relative  division  error  of  the  mean 
reading  of  the  fixed-circle  microscopes  on  the  45°  lines  is 
\  n.  Similarly,  the  45°  division  error  for  the  movable 
circle  may  be  computed. 

The  division  errors  of  the  15°  readings  of  the  two  circles 
may  be  obtained  by  subdividing  and  comparing  the  45° 
arcs  just  determined,  or  from  the  complete  circles,  as 
before ;  and  so  on  for  the  5°,  1°  and  other  readings. 

When  a  circle  has  been  investigated,  the  zero  of  the 
system  may  be  changed  arbitrarily  by  applying  a  constant 
to  all  the  division  errors  secured,  either  to  make  them  all 
of  the  same  sign,  or  to  make  their  algebraic  sum  zero. 

The  1°  readings  of  the  fixed  circle,  and  the  3°  readings 
of  the  movable  circle  of  the  Repsold  instrument  of  the 
Lick  Observatory,  were  investigated  by  the  above  methods 
by  K.  H.  Tucker.*  The  average  errors  of  the  fixed  and 

*  For  further  and  fuller  details,  see  articles  by  Professor  Tucker  in 
Publications  Astronomical  Society  of  the  Pacific,  1895,  pp.  330-338,  and 
1890,  pp.  270-272.  Also  an  article  by  Professor  Boss  in  TJie  Astronomi- 
cal Journal,  1896,  Nos.  382,  383. 


186 


PRACTICAL   ASTRONOMY 


movable  circle  readings  were  ±  0'M8  and  ±  0'M5,  re- 
spectively. The  following  table  contains  the  errors  for  the 
9°  readings,  by  way  of  illustration. 


Beading 

Fixed  Circle 

Movable  Circle 

0° 

+  0".18 

+  0".10 

9 

-1-0  .12 

+  0  .14 

18 

-0  .23 

+  0  .04 

27 

-0  .04 

+  0  .37 

36 

-0  .52 

-0  .34 

45 

-  0  .34 

+  0  .03 

54 

+  0  .02 

-0  .18 

63 

+  0  .11 

-0  .07 

72 

-0  .22 

-0  .14 

81 

+  0  .16 

+  0  .08 

90 

+  0  .18 

+  0  .10 

In  case  the  instrument  has  only  one  circle,  its  errors 
may  be  determined  by  means  of  two  extra  microscopes, 
placed  180°  apart,  in  connection  with  the  four  regular 
microscopes.* 

It  should  be  explained  that  many  observers  shift  the 
movable  circle  from  time  to  time,  so  that  the  several 
observations  of  any  star  will  depend  upon  many  different 
graduations  of  the  circle,  thereby  reducing  the  magnitude 
of  the  division  error  affecting  the  mean  result. 

The  flexure  of  the  circles  in  modern  instruments  is  so 
small  that  to  apply  a  correction  for  it  is  generally  more 
objectionable  than  to  omit  it.  This,  and  similar  small  and 
uncertain  corrections  are  not  ignored,  however.  Good 
practice  requires  that  a  star's  position  be  determined  from 
an  equal  number  of  observations  with  circle  west  and  circle 
east,  with  the  result  that  several  slight  errors  are  largely 
eliminated  from  the  mean  of  all  the  observations. 


*  For  an  exposition  of  this  method,  see  Annalen  der  Sternwarte  in 
Leiden,  Band  II,  seite  [50-92]. 


THE   MERIDIAN   CIRCLE  187 

123.  Reduction  to  the  meridian.     Theoretically,  the  ob- 
server is  supposed  to  bisect  the  image  of  a  star  with  the 
declination  micrometer  at  the  instant  of  meridian  passage. 
If  for  any  reason  the  bisection  is  made  at  t  seconds  before 
or  after  meridian  passage,  the  necessury  correction  x  to 
reduce  to  the  meridian  may  be  found  from  equation  (281) 
and  the  corresponding  table  in  §  111.     The  correction  x 
must  be  applied  to  the  circle  reading  with  the  proper  sign 
to  increase  the  observed  polar  distance. 

124.  Refraction.     The  refractions  given  by  (95)  must 
be  applied  to  the   circle  readings   in  such  a  way  as  to 
increase  the  zenith  distances. 

125.  Parallax.    Observations  on  bodies  within  the  solar 
system  will  require  correction  for  parallax,  by  the  methods 
of  §§  25-27. 

126.  The  meridian  circle  is  applied  to  the  determination 
of  declinations  by  two  general  methods. 

1st.  Fundamental  Determinations.  The  latitude  <£  of 
the  observer  being  known,  the  equator  reading  of  the  circle 

is  given  by 

Equator  reading  =  Zenith  reading  ±  <f>,  (287) 

the  lower  sign  being  for  circle  east.  The  difference 
between  the  circle  reading  for  a  star  (corrected  for  refrac- 
tion, etc.),  and  the  equator  reading,  is  the  declination  of 
the  star,  determined  fundamentally. 

2d.  Differential  Determinations.  If  a  standard  star  be 
observed  in  the  usual  manner,  its  circle  reading  (corrected 
for  refraction,  etc.),  plus  or  minus  its  known  declination, 
will  be  the  equator  reading  of  the  circle.  The  corrected 
circle  reading  for  an  undetermined  star  will  differ  from 
the  equator  reading  by  the  declination  of  the  star.  Fur- 
ther, the  equator  reading  obtained  from  the  standard  star, 
combined  with  the  zenith  reading,  will  furnish  a  new  deter- 
mination of  the  latitude. 


188  PRACTICAL   ASTRONOMY 

A  circumpolar  star  observed  for  latitude  at  both  upper 
and  lower  culmination  has  the  advantage  that  any  error 
of  declination  is  eliminated  from  the  mean  result ;  but  the 
disadvantage,  for  observers  situated  well  toward  the  equa- 
tor, that  the  refractions  are  large. 

Since  the  latitude  of  a  place  varies  appreciably,  funda- 
mental determinations  of  declination  require  a  knowledge 
of  the  current  value  of  the  latitude.  Programs  for  funda- 
mental work  should  contain  a  few  standard  stars,  as  checks 
on  each  night's  results. 

In  programs  for  differential  work,  the  undetermined 
stars  should  be  preceded,  accompanied  and  followed  by 
standard  stars ;  the  range  of  declinations  for  the  two  kinds 
being  about  equal,  to  assist  in  eliminating  uncertainties  in 
refractions.  The  equator  reading  should  be  obtained  from 
all  the  standard  stars.  A  program  covering  four  or  five 
hours  should  contain  eight  or  ten  standard  stars.  If  the 
value  of  the  latitude  is  known,  a  long  series  of  such  obser- 
vations of  the  standard  stars  will  furnish  corrections  to  the 
standard  declinations  themselves. 

The  following  abridged  program  of  observations,  made 
with  the  Repsold  meridian  circle  of  the  Lick  Observatory, 
illustrates  many  of  the  important  principles  treated  in  this 
chapter. 

The  mean  of  the  nadir  and  micrometer  readings  taken 
just  before  and  after  the  observations  of  stars  furnished  the 

values 

Nadir  reading  =  134°  56'  29".82, 

Micrometer  (zero  reading)  =  17r.OOO. 

The  meteorological  observations  required  for  computing 
the  refractions  were  made  at  15.2  hours  sidereal  time, 
thus : 

Barom.  25.70  inches,  Att.  Therm,  -f  64°  F.,  Ext.  Therm.  +  63°.0  F. 

Other  meteorological  observations  were  taken  throughout 
the  program. 


THE   MERIDIAN   CIRCLE 


189 


o 

s 

•^        rH        CO       CO 

T(H        C5        0        rH 

GN      GM 

o 

1     1 

^^     Oi 

05        0        0       TtH 
G^l        _J.         |         rH 

ut>     CO 
rH 

GN 

o     £     ^ 

S      05       —  < 
G$      0      CO 

rH       CO 

O      O 

O                                rH 
1 

^      CO 

GN 

GQ.     "O      Oi 

CO      "^ 

CO 

CO      b- 

|^_ 

o 

iO      t^ 

\ 

CO 

GN      GN 

CO 

O      O      00      G^ 

8                CO 
o 

0 

10       05 

o    o    d    co° 

^              CO 

co° 

a 

CN      GN 

^    ub    CM 

rH       CO 

O                             rH 

CO                CO 

GM 

;_,     tO      O 

"*                                 1 

Tfl                         CO 

to 

00      ^1 

CO                l~"* 

^fl      rH      GM 

iO 

O               t— 

K 

1 

CO 

GN 

1 

g 

05      0      CO     t- 
i—  (      o      O      CO 

«     s 

co 

C^l      O5 

00      O      O°      GXJ 

O                CO 

00 

25    s 

C^      CM 

1    " 

0 

^      C^      *O 

GM      >O 

Cl                                rH 

O              CO 

CO                CO 

0 

|..S> 

Jj      rH       rH 

O       rH 

s 

CO                t- 

05 

rH 

1 

CO 

G^l 

CM               C^ 

1 

o        S 

CO 

00      O      O5      O 
O      O     O     CO 

<M       rH 
O      G^ 

CO 

CO 
1      15       rH 

-*  § 

CO      CO 

CO     O     O     Tt< 

s  * 

S 

C")            ^^ 

CO      O 

W                             ^L. 

CO      CO 

GN 

KM      §>       rH 

rH       TJ4 

CO 

CO      l>- 

^ 

o    """    Hh 

CO 

* 

C<I      t^ 
<M 

CO 

r_l 

c^ 

S<J       rH       CO      CO 

t^     CO 

5 

o 

00 

CO        0       0        T|H 

C^l      00 

O5 

1  s  s  • 

CO      O5 
CO      CN 

C^       O      O       rH 
°       +         1        rH 

s  °° 

% 

k!}      O5      ">- 

CO      CO 

05                           , 

CO      CO 

O 

^  i  i 

rH       T* 

CO 

CO      t- 

^ 

rH       rH 

10  3 

S 

CO      l>~ 
GM      G<l 

CO 

CO 

eg 

CO      r-i      CO      O 

CO     O 

gq 

CO 

52                **•* 

. 

co    o    o    t- 

rH       O 

CO 

CO      O5 

CO      O      O      CO 

o    co 

CO 

^1                   "^ 

^^      GN 

rH          I           ,        CO 

to 

CM 

g      0     PH 

O      lO 

t-                + 

s  8 

o 

^  i  & 

•**      -^ 

05 

0      £- 

^ 

CO        r-  1 

CO 

CO 

CO      C^ 

CO 

• 

. 

^ 

• 

bO    .2 

. 

. 

. 

.5     "^      bo 

. 

t 

( 

§    ^   °5 

.0 

a 

'    %      ' 

'       bfi 

bD 

•5    *    '    ' 

^     °     * 

1  1  s 

's   S*  1 

11 

°3 

*     ft 

^     S3 

lill 

O     P^     PM     PH 

III 

0    0    S 

3)       C^ 

QJ          0^ 

CO       co 

,0      J3 

0    0 

190  PRACTICAL   ASTRONOMY 

By  way  of  illustration,  the  original  circle,  microscope  and 
micrometer  readings  for  the  first  star,  7  Ur&ce  Minoris,  were : 
Circle  Microscopes  *  Micrometer 

349°  46'  I    45".l       (48')  45".2  16r.500 

II    54  .0  54  .0 

III  54  .1  54  .0 

IV  55  .8  56  .0 

Means    52  .25  52  .30 

Original  circle  reading 349°  46'  52".28 

Correction  for  micrometer,  +  Or.500      .  +  24".05 

Circle  reading 349°  47'  16".33 

In  this  program,  the  first  and  third  stars  are  standard 
circumpolars,  the  former  at  upper,  and  the  latter  at  lower 
culmination.  The  second  and  sixth  stars  are  standard  stars. 
The  declinations  assigned  are  the  apparent  declinations  of 
the  standard  stars,  and  the  approximate  declinations  of  the 
undetermined  stars.  The  circle  readings  include  the  cor- 
rections for  the  readings  of  the  declination  micrometer. 
The  apparent  zenith  distances,  z,  are  the  differences  of  the 
circle  readings  and  the  zenith  reading,  to  be  used  in  com- 
puting the  refractions.  The  corrected  circle  readings, 
minus  the  declinations  of  the  standard  stars,  are  the  ob- 
served equator  readings,  and  their  mean  for  the  two  south- 
ern stars  is  adopted  as  the  equator  reading.  The  differences 
of  the  zenith  reading  and  the  observed  equator  readings 
for  the  four  standard  stars  are  four  values  of  the  observed 
latitude.  The  corrected  circle  readings  for  the  two  unde- 
termined stars,  minus  the  mean  equator  readings  are  the 
observed  declinations.  The  corrections  for  graduation 
errors  have  not  been  applied :  the  errors  for  the  particular 
graduations  used  have  not  been  determined. 

*  It  is  the  custom  of  this  observer  to  obtain  the  microscope  readings  on 
two  graduations  on  opposite  sides  of  the  micrometer  zero  positions  ;  to 
use  the  mean  of  the  seconds  given  by  the  eight  readings  ;  and  to  correct 
for  runs  for  the  distance  that  the  mean  of  the  two  graduations  is  from  the 
micrometer  zero.  The  value  of  the  correction  for  runs  is  determined  from 
all  the  readings  of  the  program. 


CHAPTER   X 
ASTRONOMICAL  AZIMUTH 

127.  In  many  problems  in  higher  surveying  the  azimuth 
of  a  point  on  the  earth's  surface,  as  viewed  from  the  point 
of  observation,  is  required  to  be  very  accurately  known. 
It  is  determined  by  measuring  the  difference  of  the  azi- 
muths of  the  point  and  a  star  by  means  of  a  theodolite, 
a  surveyor's  transit,  or  any  similar  instrument  designed 
for  measuring  horizontal  angles.  The  azimuth  of  the  star 
is  computed  from  the  known  right  ascension,  declination, 
latitude  and  time ;  whence  the  azimuth  of  the  point  can  be 
obtained. 

Only  the  four  circumpolar  stars,  whose  places  are  given 
in  the  Ephemeris,  pp.  302-313,  should  be  used  in  accurate 
determinations. 

The  point  whose  azimuth  is  to  be  determined  is  marked 
conveniently  by  a  lamp  arranged  to  shine  through  a  small 
hole  in  a  box,  placed  directly  over  the  point.  It  should  be 
at  least  one  mile  from  the  observer.  If  no  provision  has 
been  made  for  illuminating  the  wires  of  the  telescope  at 
night,  they  can  be  rendered  visible  by  tying  a  piece  of  thin 
unglazed  white  paper  over  the  object  glass  of  the  telescope, 
first  cutting  a  hole  in  the  paper  nearly  as  large  as  the 
object  glass,  and  throwing  the  light  of  a  bull's-eye  lantern 
on  the  paper. 

The  instrument  is  set  up  over  the  point  of  observation 
(marked  in  some  way)  and  carefully  adjusted.  The  hori- 
zontal graduated  circle  is  fixed  in  position  by  clamping. 
The  level  screws  and  other  adjusting  screws  must  not  be 

191 


192  PRACTICAL   ASTiiONOMY 

touched  during  a  series  of  observations.  If  the  rotation 
axis  of  the  telescope  is  not  truly  horizontal,  an  error  is 
introduced  in  the  measured  difference  of  azimuth  of  the 
mark  and  star,  which  must  be  allowed  for.  In  the  finer 
instruments  the  inclination  of  the  axis  is  measured  by 
means  of  a  striding  level.  The  effect  of  an  error  of  col- 
limation  is  practically  eliminated  by  reversing  the  instru- 
ment and  observing  an  equal  number  of  times  in  both 
positions.  We  shall  consider  that  this  is  always  done. 
If  more  than  one  series  of  observations  is  made,  the  hori- 
zontal circle  should  be  shifted  so  that  a  different  part  of 
it  may  be  used,  thereby  eliminating  largely  any  errors  of 
graduation  of  the  circle. 

128.  Correction  for  Level.  When  the  rotation  axis  of 
the  telescope  is  inclined  to  the  horizon,  the  line  of  sight 
does  not  describe  a  vertical  circle,  and  the  horizontal  circle 
reading  requires  a  small  correction.  Let  b  be  the  eleva- 
tion of  the  west  end  of  the  axis  above  the  horizon,  and  let 
the  west  end  of  the  produced  axis  cut  the  celestial  sphere 
in  TF;  let  y  be  the  corresponding  correction  to  the  circle 
reading  ;  and  let  Z  be  the  zenith  and  S  the  star.  Then,  in 
the  triangle  WZS, 


and,  therefore, 

sin  b  cos  z  —  cos  b  sin  z  sin  y  =  0. 

But  b  and  y  are  very  small,  and  we  may  write 

y  =  bcotz-,  (288) 

and  for  circumpolar  stars,  we  may  write 

y  =  b  tan  <j>.  (289) 

The  value  of  b  is  found  by  (166)  or  (167),  or  by  the 
methods  of  §  94.  If  the  illuminated  mark  is  not  in  the 
horizon,  the  circle  readings  on  the  mark  must  be  corrected 
by  (288),  using  its  zenith  distance  z. 


DETERMINATION   OF    AZIMUTH  193 

129.  Correction  for  diurnal  aberration.     Owing  to  diur- 
nal aberration  the  star  will  be  observed  too  far  east.     In 
the  most  refined  observations  this  must  be  allowed  for. 
The  correction  to  the  circle  reading  is  given  by   (118); 
which,  for  circumpolar  stars,  is  approximately 

dA  =  +  0". 31  cos  4.  (290) 

If  the  circles  cannot  be  read  to  less  than  1",  this  correc- 
tion is  negligible. 

130.  Correction  for  error  of  runs.    If  reading  microscopes 
are  used  [§  58],  the  circle  readings  must  be  further  cor- 
rected for  error  of  runs. 


AZIMUTH    BY    A    CIRCUMPOLAR    STAR    NEAR 
ELONGATION 

131.  A  star  is  at  western  or  eastern  elongation  when  its 
azimuth  is  the  least  or  greatest  possible.  It  is  then  moving 
in  a  vertical  circle,  and  is  in  the  most  favorable  position 
for  azimuth  observations.  Only  one  observation  can  be 
made  at  the  instant  of  elongation,  but  it  is  customary  to 
make  several  observations  just  before  and  after  elongation, 
and  allow  for  the  change  in  azimuth  during  the  intervals. 

At  the  instant  of  elongation,  the  triangle  formed  by  the 
pole,  star  and  zenith,  which  we  shall  denote  by  PSZ,  is 
right-angled  at  the  star.  If  we  let  #0  be  the  sidereal  time, 
and  A0,  tQ  and  20  the  azimuth,  hour  angle  and  zenith  dis- 
tance of  the  star  at  elongation,  a  and  8  its  right  ascension 
and  declination,  and  $  the  observer's  latitude,  we  shall 
have,  for  western  elongation, 

PZ=    90°-  <£,  PS  =  90°- 8,        ZS  =  zw 

PZS  =  180°  -  A  0,        ZPS  =  tv  PSZ  =  90° ; 

and  for  eastern  elongation, 

PZS  =  A0-  180°,        ZPS  =  360°  -  tv 


194  PRACTICAL  ASTRONOMY 


We  can  write 


tan  d>  sin  rf>  cos  8 


,ftni\ 

costn  =  -  —  £>        cos  zn  =  -T-^S-  >        sm^4ft=±  -  r>       (291) 
tan  8  sin  8  cos  <£ 

00  =  a  +  tv  (292) 

tQ  is  in  the  first  quadrant  for  western  elongation,  and  in  the 
fourth  for  eastern  ;  z0  is  always  in  the  first  quadrant  ;  and 
AQ  is  less  than  180°  for  western  elongation,  and  greater 
than  180°  for  eastern.  We  can  also  write 

±glnX.  =  52!«=™*^l,  (293) 

cos  <f>         sin  <£ 

±  cos  A  0  =  —  sin  8  sin  /0  ,  (294) 

in  which  the  upper  signs  are  for  western  elongation,  the 
lower  for  eastern. 

If  the  star  is  observed  at  any  other  hour  angle  £,  its 
azimuth  A  is  given  by  (16)  and  (17).  Multiplying  (16) 
by  (293),  (17)  by  (294),  and  subtracting  one  product  from 
the  other,  we  obtain 

sin  z  sin  (A  Q  -  A  )  =  =p  sin  8  cos  8  2  sin2  J  (?0  -  t)  .  (295) 

If  the  observations  are  made  near  elongation,  t  will  not 
differ  much  from  £0,  AQ  —  A  will  be  small,  and  for  the 
circumpolar  stars  z  will  not  differ  much  from  ZQ.  There- 
fore we  can  write,  without  sensible  error, 

sin^cos_S     Qsin^ftpO,  (296) 

sin  z0  sm  1" 

in  which  the  lower  sign  is  for  eastern  elongation,  as 
before.  AQ  —  A  is  the  correction  to  be  applied  to  the 
circle  reading  for  an  observation  made  at  hour  angle  £, 
to  reduce  to  the  corresponding  circle  reading  for  an  obser- 
vation made  at  hour  angle  £0. 
For  convenience,  let 


sin  1" 
and  (296)  becomes 

40  -  A  =  T  msinScosg.  (297) 

sinz0 


DETERMINATION    OF    AZIMUTH  195 

The  values  of  m  can  be  taken  from  Table  III,  Appendix, 
for  the  different  values  of  tQ  —  t.  If  we  let  m0  be  the  mean 
of  the  several  values  of  m,  the  corrections  can  be  applied 
collectively  to  the  mean  of  the  circle  readings  on  the  star, 
and  the  equation  (297)  becomes 

sin  8  cos  8 


sin  zn 


(298) 


in  which  A0  —  A  is  the  correction  to  the  mean  of  the 
circle  readings.  Further,  if  the  level  readings  have  been 
taken  symmetrically  with  reference  to  the  program,  which 
can  always  be  done,  the  mean  value  of  y,  equation  (289), 
can  be  applied  to  the  mean  of  the  circle  readings. 

132.  The  values  of  £0  and  #0  having  been  computed  for 
the  star  to  be  observed,  the  instrument  is  carefully  adjusted, 
and  a  program  similar  to  this  is  followed  : 

Make  two  readings  on  the  mark 

Read  the  level 

Make  four  readings  on  the  star 

Read  the  level 

Make  two  readings  on  the  mark 

Reverse 

Make  two  readings  on  the  mark 

Read  the  level 

Make  four  readings  on  the  star 

Read  the  level 

Make  two  readings  on  the  mark 

The  times  of  observation  are  noted  on  a  time-piece,  pref- 
erably a  sidereal  chronometer.  Its  correction  must  be 
known  within  one  or  two  seconds  if  the  most  refined  form 
of  instrument  is  employed,  or  to  the  nearest  minute  if  an 
ordinary  surveyor's  transit  is  used.  This  correction  can 
be  obtained  by  any  of  the  methods  described  in  the  pre- 
ceding chapters,  or  by  a  comparison  with  the  time  signals 
at  the  nearest  telegraph  station.  The  chronometer  time  of 
elongation  is  now  known.  Subtracting  from  it  the  several 
times  of  observation,  the  values  of  tQ  —  t  are  found,  and 


196  PRACTICAL    ASTKONOMY 

the  values  of  m  corresponding  to  them  taken  from  Table 
III.  Forming  the  mean  mQ  and  computing  z0  from  (291), 
the  value  of  A0  —  A  is  found  and  applied  to  the  mean  of 
the  circle  readings.  The  mean  of  the  corrections  for  level 
errors  and  the  correction  for  diurnal  aberration  are  now 
applied.  The  corrected  mean  circle  reading,  which  we 
shall  call  a,  corresponds  to  the  azimuth  AQ  of  the  star  at 
elongation,  which  is  computed  by  (291).  If  k  is  the  mean 
of  the  circle  readings  on  the  mark,  and  M  the  azimuth 
of  the  mark,  then 

M  =  k-(s-A0).  (299) 

The  circle  reading  when  the  telescope  is  directed  to  the 
south  point  of  the  horizon  will  be  equal  to  s  —  A0. 

133.  Example.  Detroit  Observatory,  Wednesday,  1891 
May  6.  Find  the  azimuth  of  a  given  point  (nearly  in  the 
horizon)  from  observations  on  8  Ursce  Minoris  near  its 
eastern  elongation.  Observer's  latitude,  42°  16;  48' '. 

The  apparent  place  of  the  star  was 

a  =  18*  7m  44s,  8  =  +  86°  36'  25".0. 

Equations  (291)  and  (292)  are  solved  as  below. 

tan<£  9.958704  sin<£  9.827856  cos  S  8.772214 

tan  8  1.227024  sin  8  9.999236  cos<£  9.869153 

t0  273°  5'  25"  z0  47°  37'  42"  A0  184°  35'  17".8 

t0  18»12*22« 

a  18     7  44 

00  12  20     6 

The  chronometer  correction  was  -f  18m  52s,  and,  there- 
fore, the  chronometer  time  of  elongation  was  12A1W14*. 
A  good  surveyor's  transit,  whose  horizontal  circle  was 
read  to  10"  by  two  verniers,  and  which  was  provided  with 
plate  levels  and  a  delicate  striding  level,  was  placed  over 
the  point  of  observation  and  carefully  leveled  a  short  time 
before  elongation.  The  following  observations  were  made : 


DETERMINATION    OF    AZIMUTH 


197 


No. 

Object 

Telescope 

Chronometer 

Vernier  A 

Vernier  B 

(1) 

Mark 

Reversed 

96°  16'  40" 

276°  16'  30" 

(-0 

" 

« 

96    16  35 

276    16  25 

(3) 

Level 

(4) 

Star 

" 

IP  44"  52* 

243   39  20 

63    39  20 

(5) 

M 

" 

48    40 

243    39  50 

63   39  50 

(6) 

it 

" 

51      6 

243   39  50 

63    39  50 

00 

it 

« 

53    11 

243   40     0 

63   40     5 

Level 

(9) 

Mark 

u 

96    16  45 

276    16  40 

(10) 

" 

M 

96    16  50 

276    16  40 

(U) 

u 

Direct 

276    17     0 

96    16  45 

(12) 

u 

"    ' 

276    16  55 

96    16  45 

(13) 

Level 

(14) 

Star 

M 

12     5    50 

63    40  10 

243   40  10 

(15) 

M 

" 

7    54 

63    40     0 

243   40     0 

(16) 

" 

M 

9    44 

63    39  50 

243    39  50 

(17) 

" 

a 

11    27 

63    39  45 

243    39  55 

(18) 

Level 

(19) 

Mark 

« 

276    16  45 

96    16  30 

(20) 

M 

u 

276    16  45 

96    16  35 

The  level  readings  given  by  the  striding  level  were 


(3) 

W     E 
4.4    4.1 

4.3  4.0 

4.4  4.0 
4.3    3.8 


(8) 
W     E 

4.2  4.0 

4.3  3.8 
4.2    3.9 

4.4  3.8 


(13) 
W      E 
4.0    4.4 
4.0    4.2 

4.0  4.2 

4.1  4.4 


(18) 
W      E 

4.0  4.2 

4.1  4.2 
4.0    4.2 

4.2  4.2 


The  value  of  one  division  of  the  level  was  10". 7,  and 
therefore,  from  (166),  the  inequality  of  the  pivots  being 
negligible, 

(3)  (8)  (13)  (18) 

b=  +  2".0,         +  2".l,         -  1".4,         -  0".6, 


and  by  (289), 

(3) 

y  =  + 1".8, 


(8) 


(13) 


(18) 
-  0".6. 


198 


PRACTICAL   ASTRONOMY 


The  solution  of  (298)  for  the  eight  readings  on  the  star 
is  given  below.  The  column  "  Circle  Readings  "  is  formed 
by  taking  the  means  of  Vernier  A  and  Vernier  B. 


No, 

Circle  Readings 

to-t 

m 

(4) 

243°  39'  20" 

+  16™  22« 

525".7 

(5) 

243  39  50 

+  12  34 

310  .0 

(6) 

243  39  50 

-j-10   8 

201  .6 

(7) 

243  40  2 

-f  8   3 

127  .2 

(14) 

63  40  10 

-  4  36 

41  .5 

(15) 

63  40  0 

-  6  40 

87  .3 

(16) 

63  39  50 

-  8  30 

141  .8 

(17)  * 

63  39  50 

-10  13 

204  .9 

Means 


63   39  51.5 


m0  =  205  .0 


Iogm0 

sin  8 

cos  8 

cosec  z0 


A0-A 


2.31175 
9.99924 
8.77221 
0.13148 
1.21468 
16".4 


The  mean  of  the  four  values  of  y  is  4-  0".4.  The  value 
of  dA,  from  (290),  is  -  0".3.  The  corrected  circle  read- 
ing on  the  star  at  elongation  is  therefore 

5  =  63°  39'  51".5  +  16".4  +  0".4  -  0".3  =  63°  40'  8".0. 
The  mean  of  all  the  readings  on  the  mark  is 

£  =  276°16'41".6; 
and,  therefore,  by  (299), 


Since  the  verniers  on  this  instrument  read  to  only  10", 
the  diurnal  aberration  could  have  been  neglected,  and  the 
other  corrections  computed  to  the  nearest  second  only. 
But  all  the  corrections  have  been  applied  here,  to  illustrate 
the  method. 


DETERMINATION    OF    AZIMUTH  199 

AZIMUTH   BY   POLARIS   OBSERVED   AT  ANY   HOUR   ANGLE 

134.  When  the  azimuth  is  required  with  the  greatest 
possible  accuracy,  the  observations  should  always  be  made 
at  or  near  elongation,  and  reduced  as  in  the  preceding  sec- 
tion. However,  good  results  can  be  obtained  by  observing 
Polaris  in  any  position,  if  the  time  is  accurately  known. 
The  time  should  be  known  within  (K5  when  using  the 
finest  instruments,  and  within  5s  or  10s  when  using  a  good 
surveyor's  transit  whose  least  reading  is  10". 

As  in  the  preceding  method,  the  observations  should  be 
made  on  the  mark  and  star  in  both  positions  of  the  tele- 
scope. If  the  observations  are  made  in  quick  succession, 
the  mean  of  two  or  three  observations  made  before  revers- 
ing may  be  treated  as  a  single  observation,  and  similarly 
for  those  made  after  reversing.  But  if  the  separate  obser- 
vations are  several  minutes  apart,  each  observation  should 
be  reduced  separately. 

The  sidereal  time  0  of  observation  having  been  noted 
with  great  care,  the  hour  angle  t  of  Polaris  is  given  by  (41). 
If  we  let  the  azimuth  A  of  the  star  be  measured  from  the 
north  point,  4-  if  the  star  is  west  of  the  meridian  and  —  if 
east,  and  let  q  be  the  star's  parallactic  angle  [§  6],  we  may 
write  [Chauvenet's  Sph.  Trig.,  §  24] 

tan  \  (q  +  A)  =  ^  ?  L  ~  t] cot  \  t  =/cot  $  t,  (300) 

The  auxiliary  quantities,/  and/ ',  depending  on  B  and  </>, 
are  constant  for  a  night's  observations,  and  with  surveyors' 
instruments  may  be  considered  constant  for  several  weeks. 
When  they  have  been  computed,  once  for  the  whole  series 
of  observations,  they  may  be  combined  rapidly  with  the 


200 


PRACTICAL   ASTRONOMY 


values  of  cot  J  t  for  the  individual  observations,  to  deter- 
mine %(q  +  A)  and  \  (<?  —  J.),  and  thence  A  by  (302). 
The  values  of  q  need  not  be  determined  at  all.  The  cor- 
rection for  level  is  given,  as  before,  by  (288),  and  for 
diurnal  aberration  by  (290). 

With  A  representing  the  azimuth  of  the  star  measured 
from  the  north  point  as  defined  above,  and  computed  by 
means  of  (300),  (301)  and  (302),  let  8  be  the  circle  read- 
ing on  the  star  corrected  for  level  and  aberration,  K  the 
mean  of  all  the  readings  on  the  mark,  and  N  the  azimuth 
of  the  mark  measured  from  the  north  point,  +  if  west  of 
north,  and  —  if  east.  Then,  assuming  that  the  circle  read- 
ings increase  in  the  direction  of  motion  of  the  hands  of  a 
watch,  we  shall  have 


A  - 


(303) 


and  S  +  A  will  be  the  circle  reading  when  the  instrument 
points  due  north. 

Example.  Detroit  Observatory,  Wednesday,  1891  May  6. 
Find  the  azimuth  of  a  given  point  nearly  in  the  horizon, 
from  the  following  observations  of  Polaris,  made  with  the 
instrument  described  in  §  133. 


No. 

Object 

Telescope 

Chronometer 

Vernier  A 

Vernier  B 

(1) 

Mark 

Direct 

276°  16'  40" 

96°  16'  35" 

(2) 

u 

u 

276    16    40 

96    16    30 

(3) 

Level 

(4) 

Star 

H 

13*  22"*  59' 

59    15    50 

239    15   40 

(5) 

u 

u 

13  26  30 

59    17      0 

239    16    50 

(6) 

u 

u 

13  27  57 

59    17    30 

239    17    20 

(7) 

st 

Reversed 

13  32  47 

239    19    40 

59    19    45 

(8) 

it 

« 

13  34   56 

239    20    35 

59    20    30 

(9) 

u 

« 

13  36  40 

239    21    20 

59    21    15 

(10) 

Level 

(11) 

Mark 

« 

96    16    40 

276    16    30 

(12) 

« 

« 

96    16    40 

276    16    35 

DETERMINATION   OP  AZIMUTH  201 

The  striding  level  gave 

(3)  (10) 

WE  WE 

3.6    5.3  4.9    4.1 

4.6    4.3  4.5    4.5 

whence,  by  (166), 

(3)  (10) 

b  =  -  3".3,  +  2".l. 

The  position  of  Polaris  was,  American  Ephemeris,  p.  306, 

a  =  1*  17™  53%  8  =  88°  43'  29", 

and  the  chronometer  correction  was  +  18m  52*. 

The  means  of  the  observations  made  before  and  after 
reversal  are  reduced  below. 


Before 

After 

Chronometer 

13*  25™  49- 

13*  34»»  48* 

A0 

+  18    52 

+  18    52 

0 

13   44    41 

13   53    40 

a 

1   17    53 

1    17    53 

0-a  =  t 

12  26    48 

12   35    47 

t 

186°  42'     0" 

188°  56'  45" 

\* 

93   21      0 

94   28    22 

8 

+    88  43    29 

* 

+    42   16    48 

8  -  <f> 

+    46   26    41 

8+<£ 

+  131     0    17 

1(8-*) 

+    23   13    20 

1  (8  +  *) 

+    65   30      8 

cos  £  (8  -  <f>) 

9.963307 

sin£(8  +  <£) 

9.959031 

log/ 

0.004276 

0.004276 

cot  %t 

8.767417n 

8.893338M 

\(q  +  A) 

-  3°  22'  59" 

-4°  31'    1" 

sin  $(8  —  *) 

9.595825 

cos  i  (8  +  <£) 

9.617690 

log/' 

9.978135 

9.978135 

cot£* 

8.767417n 

8.893338n 

202  PRACTICAL   ASTRONOMY 

\(q-A)  -3°  11'    9"  -4°  15'  14" 

A  -  0   11  50  -  0   15  47 

Circle  on  star  59°  16'  42"  239°  20'  31" 

y  -  3  4-2 

dA  0  0 

S  59   16  39  239   20  33 

S  +  A  59     4  49  239     4  46 

K  276   16  36  96   16  36 

N  142   48  13  142  48  10 

Mean  N  142°  48'  12" 

The  corrections  should  be  carried  to  tenths  of  seconds 
when  refined  instruments  are  used. 

The  azimuth  of  the  same  mark  was  measured  on  the 
same  night  with  the  same  instrument,  by  means  of  obser- 
vations of  8  Ursce  Minoris  taken  near  eastern  elongation 
[see  example  of  the  preceding  section].  The  azimuth 
obtained,  measured  from  the  south  point,  was 
M  =  37°  11'  51" A. 


CHAPTER   XI 
THE   SURVEYOR'S  TRANSIT 

135.  The  surveyor's  transit  is  adapted  to  the  determina- 
tion of  the  time,  latitude  and  azimuth  by  many  of  the 
preceding  methods.     These  elements  can  easily  be  deter- 
mined to  an  accuracy  within  the  least  readings  of  the  cir- 
cles, if  the  instrument  is  of  reliable  make,  and  is  provided 
with  spirit  levels.     We  shall    assume  that  the  observer 
uses  a  mean  time-piece,  which  we  shall  call  a  watch,  and 
that  he  has  a  thorough  knowledge  of  the  subject  of  TIME, 
CHAPTER  II,  without  which  the  Ephemeris  cannot  be  used 
intelligently.      We   shall   assume,  also,  that   the    vertical 
circle  of  his  instrument  is  complete,  and  that  the  degrees 
are  numbered  consecutively  from  0  to  360.     In  case  they 
are  not,  the  observer  can  readily  reduce  his  readings  to 
that  system.     The  instrument  is  supposed  to  be  carefully 
adjusted.     A  method  of  illuminating  the  wires  at  night 
is  given  in  §  127. 

Figure  26  illustrates  a  form  of  instrument  well  adapted 
to  the  solution  of  the  problems  described  in  this  chapter ; 
but  the  methods  can  be  used,  within  limits,  with  nearly 
all  forms  of  the  surveyor's  transit  instrument. 

DETERMINATION  OF  TIME 

136.  By  equal  altitudes  of  a  star.     Set  the  instrument 
up  firmly,  level  it,  and  direct  the  telescope  to  a  known 
bright  star  east  of  the  meridian.     Pointing  the  telescope 
slightly  above  the  star,  clamp  the  vertical  circle  and  note 
the  time   T'  when  the  star  crosses   the   horizontal  wire. 

203 


FIG 


DETERMINATION   OF   TIME  205 

The  vertical  circle  must  not  be  undamped.  A  short  time 
before  the  star  reaches  the  same  altitude  west  of  the 
meridian,  level  the  instrument,  move  it  in  azimuth  until 
the  telescope  is  directed  to  a  point  just  below  the  star, 
wait  for  the  star  to  enter  the  field,  and  note  the  time  T" 
when  it  crosses  the  horizontal  wire.  The  sidereal  time  6 
when  the  star  was  on  the  observer's  meridian  equals  its 
right  ascension  a,  and  this  corresponds  to  the  mean  of  the 
two  watch  times.  Converting  the  sidereal  time  6  =  a  into 
the  corresponding  mean  time  T,  the  watch  correction  AT 

is  given  by 

AT"  =  T  -  %(T'  +  T").  (304) 


Example.  Thursday,  1891  March  5.  In  longitude 
5/t  34™  55^  Regulus  was  observed  at  equal  altitudes  east 
and  west  of  the  meridian,  at  the  watch  times 

T  =  8*  7m  34%  T"  =  14*  10™  20«. 

Required  the  watch  correction. 

From  the  American  Ephemeris,  p.  332,  a=0=lQh  2m  355. 
Converting  this  into  mean  time,  §  18,  we  find 


and,  therefore,  by  (304),  the  watch  correction  was 

AT  =  -54*; 

or  the  watch  was  54s  fast. 

137.  By  a  single  altitude  of  a  star.  Level  the  transit 
instrument.  Direct  the  telescope  very  slightly  above  a 
known  star  in  the  east  or  below  a  known  star  in  the 
west,  and  clamp  the  telescope.  Note  the  watch  time  when 
the  star  crosses  the  horizontal  wire  and  read  the  vertical 
circle.  Unclamp  the  telescope  and  repeat  the  observation 
once  or  twice,  as  quickly  as  possible.  Double  reverse  the 
instrument,  and  make  the  same  number  of  observations  as 
before.  Form  the  means  of  the  circle  readings  made  be- 


206  PRACTICAL  ASTRONOMY 

fore  reversal  and  of  those  made  after.  Subtract  one  from 
the  other  in  that  order  which  makes  their  difference  less 
than  180°.  One  half  this  difference  is  the  apparent  zenith 
distance  of  the  star  at  T',  the  mean  of  the  several  watch 
times  of  observation.  Adding  the  refraction  given  by 
(97), 

r  =  58"  tan  z,  (305) 

the  result  is  the  true  zenith  distance  z.  Substituting  the 
values  of  z,  <£  and  8  in  (38)  or  (39),  the  hour  angle  t  is 
found.  The  sidereal  time  0  is  given  by  [§  8], 

0  =  a  +  t.  (306) 

Converting  0  into  the  mean  time  T,  the  watch  correction 

is  given  by 

A!T=  T-  T'.  (307) 

Example.  Saturday,  1891  April  25.  In  latitude 
+  42°  16'  47"  and  longitude  5*  34W  55s,  the  following  alti- 
tudes of  Arcturus  were  observed  east  of  the  meridian. 
Find  the  watch  correction. 

Telescope  Watch  Circle  reading 

Direct  7*52»23«                 34°  43'   0" 

«  53    33                  34  55  30 

«  54   20                  35     4    0 

Reversed  55   37  144  42  30 

«  56   30  144  33    0 

«  57    18  144  24    0 

The  means  of  the  circle  readings  are  34°  54'  10"  and 
144°  33'  10";  and  one  half  their  difference  is  the 

Apparent  zenith  distance,  54°  49'  30"          log  58  1.7634 

Refraction,  r,           122             tanz  0.1520 

True  zenith  distance,  z,  54  50  52              logr  1.9154 

r  82" 

From  the  Ephemeris,  p.  340, 

a  =  14*  10OT  43«,         8  =  +  19°  44'  52". 


DETERMINATION    OP   TIME  207 

The  solution  of  (38)  is 

z       54°  50'  52"  log  sin  \  \z  +  (<£  -  8)]  9.79595 

<j>  +  42  16  47  log  sin  I  [z  -  (<£  -  8)]  9.44449 

8  +  19  44  52  log  sec  \  [z  +(<£  +  8)]  0.28114 

<£  -  8       22  31  55  log  sec  \\z  -  (<f>  +  8)]  0.00085 

<£  +  8       62     1  39  log  tan2  \t  9.52243 

z  +  (<£  -  8)       77  22  47  log  tan  \t  9.76121rt 

2  -(<£  -  8)       32  IS  57  ^  150°  0'  48" 

z  +  (<£  +  8)      116  52  31  <  300  1  36 

z  -(<£  +  8)   --    7  10  47  *      20*0™  6s 

Solving  (306),  6  =  10*  Wm  49s.  The  equivalent  mean 
time  is  T=  7^55"*  45s;  and  the  mean  of  the  six  times  of 
observation  is  T'  =  7*  55™  2s.  Therefore, 


138.  J?y  a  single  altitude  of  the  sun.*  Observe  the 
transits  of  the  sun's  upper  and  lower  limbs  over  the  hori- 
zontal wire  by  the  method  used  for  a  star,  §  137.  Double 
reverse,  and  repeat  the  observations.  f  Form  half  the 
difference  of  the  means  of  the  circle  readings,  and  add  the 
refraction  given  by  (305),  as  before.  Further,  subtract 
the  parallax  given  by  (64) 

;>  =  9"sinz,  (308) 

and  the  result  is  the  sun's  true  zenith  distance  z  at  the 
mean  of  the  times,  T'.  The  correct  mean  time  is  probably 
known  within  5W  or  10m.  Increase  it  by  the  longitude, 
and  the  result  is  an  approximate  value  of  the  Greenwich 
mean  time.  Take  from  the  Ephemeris,  p.  II  of  the  month, 
the  value  of  the  sun's  declination  8  at  that  time.  The. 

*  The  observer  must  cover  the  eyepiece  with  a  small  piece  of  very 
dense  neutral-tint  glass  before  looking  through  at  the  sun.  The  observa- 
tions can  be  made,  also,  by  holding  a  piece  of  paper  a  short  distance  back 
from  the  eyepiece,  and  focusing  the  eyepiece  so  that  the  images  of  the 
sun  and  wire  are  seen  on  the  paper. 

t  While  waiting  for  the  second  limb  to  approach  the  wire,  the  time 
may  well  be  spent  in  reading  the  vertical  circle. 


Limb 

Watch 

Circle  reading 

Upper 

20*  38™  59' 

41°  48'  30" 

Lower 

41    59 

41   48  30 

Upper 

46   49 

137  33    0 

Lower 

49    50 

137  33    0 

208  PRACTICAL  ASTRONOMY 

Ephemeris  contains  the  apparent  declination  for  Greenwich 
mean  noon,  and  the  "  difference  for  one  hour,"  whence  the 
declination  at  any  instant  can  be  found.  Solve  (38)  or 
(39)  for  these  values  of  z,  £  and  <f>.  The  resulting  hour 
angle  t  is  the  observer's  apparent  solar  time.  Convert 
this  into  the  equivalent  mean  time  T,  by  §  15.  The  watch 
correction  is  given  by  (307),  as  before. 

Example.  Thursday  morning,  1891  May  6.  In  latitude 
+,42°  16'  50"  and  longitude  5*  35m,  the  following  observa- 
tions of  the  sun  were  made  with  Buff  &  Berger  transit  No. 
1554.  Required  the  watch  correction. 

Telescope 

Direct 
tt 

Reversed 


One  half  the  difference  of  the  circle  readings  is 
47°  52'  15".  The  refraction,  by  (305),  is  64",  and  the 
parallax,  by  (308),  is  7".  Therefore  the  true  zenith  dis- 
tance z  of  the  sun's  center  is  47°  53'  12".  The  mean  of 
the  four  watch  times  is  20*  44m  24s.  We  have 

T'  1891  May  6*20*  44™  24- 
Longitude  5  35 

Gr.  mean  time  7     2  19 

"        "        "  7     2A.32 

From  the  American  Ephemeris,  p.  75,  the  sun's  declina- 
tion at  Greenwich  mean  noon,  May  7,  was  4- 16°  48'  53", 
and  the  difference  for  one  hour,  -f-  41".  The  change  for 
2*.32  was  therefore  96",  and  the  required  value  of  the 
declination  was  8  =  -f- 16°  50'  29". 

Substituting  the  values  of  z,  <f>  and  8  in  (38),  and  solv- 
ing as  was  done  in  §  137,  we  obtain  the  hour  angle  t  = 
312°  12'  10"  =  20A  48™  49s.  The  observer's  true  time  is 
therefore  May  6rf  20*  48m  49s.  Converting  this  into  the 


DETERMINATION   OF   LATITUDE  209 

mean  time  T,  by  §  15,  we  find  T=  1891  May  Qd  20A  45m  15*. 
The  watch  correction  is 

AT  =  20  45*  15'  -  20*  44"  24«  =  +  51«. 


DETERMINATION   OF   GEOGRAPHICAL   LATITUDE 

139.  By  a  meridian  altitude  of  a  star.  A  star  is  on  the 
observer's  meridian  when  the  sidereal  time  6  is  equal  to 
its  right  ascension  a.  Convert  this  into  the  corresponding 
mean  time,  subtract  the  watch  correction  obtained  by  any 
of  the  above  methods  from  it,  and  the  result  is  the  watch 
time  of  the  star's  meridian  passage.  A  few  seconds  before 
this  watch  time  direct  the  telescope  to  the  star,  bring  the 
star's  image  on  the  horizontal  wire,  and  read  the  circle. 
Double  reverse  quickly,  and  make  another  observation. 
As  before,  form  one  half  the  difference  of  the  circle  read- 
ings, add  the  refraction  given  by  (305),  and  the  sum  is  the 
star's  true  zenith  distance  z.  Take  the  value  of  8  from 
the  Ephemeris.  For  a  star  observed  south  of  the  zenith, 

<j>  =  8  +  z;  (309) 

and  for  a  star  observed  between  the  zenith  and  pole, 

<£  =  8  -  z.  (310) 

For  a  star  below  the  pole  the  sidereal  time  of  meridian 
passage  is  12^  +  a.  Obtaining  the  value  of  z  as  before,  the 

latitude  is  given  by 

4>  =  180°-S-z.  (311) 

Example.  Ann  Arbor,  Friday,  1891  April  24.  a  Hydras 
was  observed  on  the  meridian  with  a  surveyor's  transit,  as 
below.  Required  the  latitude. 

Telescope        Circle  reading 
Reversed  140°  26'  30" 

Direct  39   33     0 

One  half  the  difference  of  the  circle  readings  is  50°  26' 
•45".  The  refraction  is  70".  Therefore,  z  =  50°  27'  55". 


210  PRACTICAL    ASTRONOMY 

From  the  Ephemeris,  p.  331,  8  =  -  8°  11'  18".     Therefore, 

from  (309), 

4  =  42°  16'  37". 

To  find  the  watch  time  when  the  star  is  on  the  meridian, 
we  have,  from  the  Ephemeris,  a  =  0  =  9*  22m  14*.  The 
corresponding  mean  time,  by  §  18,  is  lh  llm  135.  The 
watch  correction  is  4-  435,  whence  the  required  watch  time 
is  lh  10m  30s. 

140.  By  a  meridian  altitude  of  the  sun.  The  sun  is  on 
the  meridian  at  the  apparent  time  Oh  Om  O5.  Apply  the 
equation  of  time  to  this,  by  §  15,  and  subtract  the  known 
watch  correction.  The  result  is  the  watch  time  of  the 
sun's  meridian  passage.  One  or  two  minutes  before  this 
watch  time,  direct  the  horizontal  wire  of  the  telescope  to 
the  upper  limb  of  the  sun,  and  read  the  vertical  circle. 
Observe  the  lower  limb  in  the  same  way.  Double  reverse 
and  observe  both  limbs  again.  Form  half  the  difference  of 
the  means  of  the  readings  in  the  two  positions.  Add  the 
refraction  given  by  (305),  and  subtract  the  parallax  given 
by  (308).  The  result  is  the  value  of  z.  Take  from  the 
Ephemeris  the  value  of  8  for  the  time  of  meridian  passage. 
The  latitude  is  now  given  by  (309),  as  in  the  case  of  a 
star. 

Example.  Wednesday,  1891  March  25.  In  longitude 
5*  35m  the  following  meridian  altitude  observations  of  the 
sun  were  made  with  a  surveyor's  transit.  Required  the 

latitude. 

Telescope  Limb  Circle  reading 

Direct  Upper  49°  54'  30" 

"  Lower  49   22  30 

Reversed  «  130     5  30 

«  Upper  130   38  30 

One  half  the  difference  of  the  means  of  the  circle  read 
ings  is  40°  21'  45".  The  refraction  is  49".  The  parallay 
is  6".  Therefore,  z  =  40°  22'  28". 


DETERMINATION   OF   AZIMUTH  211 

The  Greenwich  apparent  time  of  observation  was  March 
25rf  5*  35m.  The  value  of  B  at  that  instant  was  4-  1°  54'  32", 
Ephemeris,  p.  38.  Therefore,  by  (309), 

</>  =  42°  17'  0". 
To  find  the  watch  time  of  meridian  passage,  we  have, 

Apparent  time  0*  Ow  0« 

Equation  of  time  +6     3 

Mean  time  063 

Watch  correction  —  0  15 

Watch  time  0  6  18 

DETERMINATION   OF   AZIMUTH 

141.  The  two  methods  of  determining  azimuth  which 
are  described  in  the  preceding  chapter  are  adapted  to  the 
surveyor's  transit,  and  need  no  further  explanation.  With 
this  instrument  the  diurnal  aberration  can  be  neglected. 

If  the  transit  is  provided  with  plate  levels  only,  they 
should  be  kept  in  perfect  adjustment.  If  the  bubble  of 
the  level  which  is  parallel  to  the  rotation  axis  of  the  tele- 
scope remains  constantly  in  the  center,  no  correction  for 
level  is  required.  But  if  the  bubble  is  n  divisions  of 
the  level  from  the  center  when  an  observation  on  a  star 
is  made,  and  d  is  the  value  of  one  division  of  the  level, 
the  circle  reading  must  be  corrected  by 

y=±wdcotz;  (312) 

H-  if  the  bubble  is  too  far  west,  —  if  too  far  east. 


CHAPTER  XII 
THE   EQUATORIAL 

142.  This  instrument  consists  essentially  of  the  follow- 
ing parts :  A  supporting  pier ;  a  polar  axis  parallel  to  the 
earth's  axis,  supported  at  two  or  more  points  by  the  pier  in 
such  a  way  that  it  can  rotate ;  a  declination  axis  attached 
to  the  upper  end  of  the  polar  axis,  and  at  right  angles 
to  it,  in  such  a  way  that  it  can  rotate ;  a  telescope  firmly 
attached  to  one  end  of  the  declination  axis,  and  at  right 
angles  to  it ;  a  graduated  declination  circle  attached  to  the 
other  end  of  the  declination  axis ;  a  graduated  hour  circle 
attached  to  the  polar  axis,  and  at  right  angles  to  it;  a 
finding  telescope  or  finder,  to  assist  in  pointing  the  principal 
telescope,  and  attached  to  it ;  a  driving  clock  and  train  of 
wheels  for  rotating  the  instrument  about  its  polar  axis  at  a 
uniform  rate.  The  various  moving  parts  are  so  counter- 
poised that  the  telescope  will  be  in  equilibrium  in  all  posi- 
tions. The  principal  features  of  the  equatorial  are  well 
illustrated  in  Fig.  27. 

A  sidereal  chronometer  is  an  almost  indispensable  com- 
panion of  the  equatorial. 

The  equatorial  serves  two  purposes : 

1st.  As  an  instrument  of  direct  observation  and  dis- 
covery, by  assisting  the  vision. 

2d.  As  an  instrument  for  determining  very  accurately 
the  relative  positions  of  two  objects  comparatively  near 
each  other,  by  means  of  a  micrometer  eyepiece  [§  60].  If 
the  position  of  one  of  the  objects  is  known,  the  position  t>f 

212 


THE   EQUATORIAL 


213 


the  other  is  known  as  soon  as  their  relative  positions  are 
determined. 


FIG.  27 


143.    By  the  above  system  of  mounting  it  is  evident  that 
the  telescope  can  be  directed  to  any  part  of  the  sky ;  and 


214  PRACTICAL   ASTRONOMY 

that  it  will  follow  a  star  in  its  diurnal  motion,  by  revolv- 
ing the  instrument  about  the  polar  axis  alone ;  for  in  that 
case  the  line  of  sight  maintains  a  constant  angle  with  the 
celestial  equator,  and  therefore  describes  a  circle  which  is 
identical  with  the  star's  diurnal  circle.  Since  the  star's 
angular  motion  is  uniform,  the  telescope  is  made  to  follow 
it  by  means  of  the  sidereal  driving  clock.  In  some  obser- 
vations the  driving  clock  is  not  used;  in  others  it  is 
indispensable. 

When  the  telescope  is  revolved  upon  the  declination 
axis,  its  line  of  sight  describes  an  hour  circle  on  the  celestial 
sphere.  The  position  of  this  hour  circle  is  indicated  by 
.the  reading  of  the  graduated  hour  cirole  of  the  instrument. 
The  position  of  the  telescope  in  this  hour  circle  is  indicated 
by  the  reading  of  the  graduated  declination  circle.  When 
the  telescope  is  directed  exactly  to  the  south  point  of  the 
equator,  the  hour  circle  reading  should  be  Oft  Om  0s,  and  the 
declination  circle  reading  should  be  0°  0'  0".  Then,  if  the 
other  parts  of  the  instrument  are  in  adjustment,  and  the 
telescope  is  directed  to  a  star,  the  hour  angle  and  declina- 
tion of  the  star  will  be  indicated  (neglecting  the  refraction 
and  parallax),  by  the  hour  circle  and  declination  circle 
readings.* 

ADJUSTMENTS 

144.  It  is  not  essential  that  the  errors  of  adjustment  of 
an  equatorial  be  entirely  eliminated,  or  that  their  values 
be  accurately  known ;  but  it  is  a  practical  convenience  to 
have  the  errors  small,  particularly  so  for  observations  on 
objects  near  the  poles  of  the  equator. 

It  is  expected  that  the  maker  of  the  instrument  will 

*  The  hour  circle  should  read  time.  It  should  be  graduated  from  0*  to 
24*  toward  the  west,  or  from  0*  to  12*  in  both  directions  from  0*.  The 
declination  circle  will  read  arc.  It  may  be  graduated  from  0°  to  360°,  or 
from  0°  to  180°  in  both  directions  from  one  of  its  equator  points,  or  from 
0°  to  90°  in  both  directions  from  its  two  equator  points.  We  shall  suppose 
it  to  read  from  0°  to  360°. 


ADJUSTMENTS  215 

adjust  the  various  parts  of  it  as  perfectly  as  possible  with 
reference  to  each  other.  It  remains  for  the  observer  to  place 
the  instrument  as  a  whole  in  correct  position. 

The  polar  axis  should  be  in  the  plane  of  the  meridian  ; 
the  elevation  of  the  polar  axis  should  equal  the  latitude  of 
the  place ;  the  hour  circle  should  read  zero  when  the 
telescope  is  in  the  meridian ;  the  declination  circle  should 
read  zero  when  the  telescope  is  in  the  equator;  and  the 
lines  of  sight  of  the  finder  and  telescope  should  be  parallel. 

The  instrument  should  first  be  placed  as  nearly  as  possible 
in  position,  by  estimation.  Then  direct  the  telescope  to  any 
known  star  near  the  southern  horizon  whose  right  ascension 
is  a.  The  star  will  be  on  the  meridian  at  the  sidereal  time 
6  =  a.  Move  the  whole  instrument  in  azimuth  so  that  the 
star  is  in  the  center  of  the  telescope  when  the  chronometer 
time  plus  the  chronometer  correction  is  equal  to  6  =  a. 
The  order  of  making  the  final  adjustments  is  as  below. 

145.  To  adjust  the  finder.     Using  the  lowest  power  eye- 
piece, direct  the  telescope  to  a  bright  star.     Replace  the 
low-power  eyepiece  by  a  high-power.     Keeping  the  star  in 
the  center  of  the  field  of  view,  turn  the  adjusting  screws  of 
the  finder  so  that  the  star  is  on  the  intersection  of  the  cross 
wires  in  the  finder.      The  two   telescopes  will   then  be 
sufficiently  near  parallelism. 

146.  To  determine  the  angle  of  elevation  of  the  polar  axis, 
and  the  index  correction  of  the  hour  circle.*    Across  the 
object  end  of  the  telescope  firmly  tie  a  piece  of  wood  which 
projects  several  inches  from  the  telescope  tube  on  the  side 
opposite  the  pier.     Pass  a  fine  thread  through  a  very  small 
hole  in  the  projecting  end,  and  fasten  it.     Direct  the  tele- 
scope to  the  zenith.     Near  the  eye  end  and  on  the  same 

*  This  very  simple  and  satisfactory  method  was  proposed  by  Professor 
Schaeberle,  in  der  Astronomische  Nachrichten,  No.  2374.  It  has  the 
advantage  that  the  errors  can  be  determined,  and  corrected,  in  the  day- 
time. 


216  PRACTICAL  ASTRONOMY 

side  as  the  projecting  arm,  fasten  a  block  of  wood.  To 
this  screw  a  metal  plate  so  that  it  will  be  perpendicular 
to  the  axis  of  the  tube,  and  in  which  is  a  very  small  circu- 
lar hole  as  nearly  as  possible  (by  estimation)  under  the 
hole  above.  Pass  the  thread  through  it,  tie  a  plumb-bob 
to  the  end  of  the  thread  near  the  floor,  and  let  it  swing  in 
a  vessel  of  water.  Move  the  telescope  by  the  slow-motion 
screws  until  the  plumb-line  passes  through  the  center  of 
the  lower  hole.  Read  both  verniers  of  the  hour  and  decli- 
nation circles.  Unclamp,  hold  the  plumb-bob  in  the  hand 
to  avoid  displacing  the  metal  plate,  reverse  the  telescope 
to  the  other  side  of  the  pier,  and  set  it  so  that  the  plumb- 
line  again  passes  centrally  through  the  hole.  Read  both 
circles  as  before. 

Let  h  equal  the  angle  of  elevation  of  the  polar  axis. 
The  difference  of  the  readings  of  the  declination  circle  in 
the  two  positions  is  180°—  2  h.  The  elevation  should  equal 
the  known  latitude  <£.  The  error  is  h  —  (f>.  Change  the 
last  circle  reading  by  this  amount,  by  moving  the  telescope 
in  declination  in  the  proper  direction.  Adjust  the  angle 
of  elevation  by  the  proper  screws  until  the  plumb-line 
again  passes  through  the  center  of  the  hole. 

If  the  declination  circle  is  graduated  so  as  to  read  from 
0°  to  90°  in  both  directions  from  its  two  equator  points, 
then  the  mean  of  the  circle  readings  for  the  two  positions 
of  the  telescope  is  at  once  the  inclination  of  the  polar  axis 
to  the  horizon. 

The  mean  of  the  hour  circle  readings  in  the  two  posi- 
tions is  the  reading  of  the  circle  when  the  telescope  is  in 
the  meridian.  This  should  be  Qh  Om  0s.  The  index  error 
of  the  hour  circle  is  the  mean  of  the  readings  minus  0^  Om  0s 
(or  minus  24^  Om  O5).  To  correct  for  it,  set  the  circle  at 
this  mean  reading ;  then  move  the  vernier  screws  until  the 
reading  is  0"  Om  0s. 

The  index  correction  of  the  hour  circle  is  equal  to  the 
index  error  with  its  sign  changed.  If  the  error  is  not 


ADJUSTMENTS 


217 


removed  by  adjusting  the  verniers,  the  index  correction 
must  be  applied  to  every  reading  made  with  the  hour  circle, 
in  order  to  obtain  the  true  reading. 

If  the  errors  are  large,  these  adjustments  should  be 
repeated  once  or  twice. 

Example.  1891  Feb.  20.  The  following  plumb-line 
observations  were  made  on  the  6-inch  equatorial  of  the 
Detroit  Observatory.  Determine  the  errors.  The  last 
column  gives  the  position  of  the  telescope  with  reference 
to  the  pier: 


HOUR  CIRCLE 

DECLINATION  CIRCLE 

TELESCOPE 

Vernier  A 

Vernier  B 

Vernier  A 

Vernier  B 

24ft     2™  53« 
11    56    56 

12*     2«  58s 
23    57      6 

135°  45'  30" 
40   16  45 

315°  45'  00" 
220    16   30 

E 

w 

The  means  of  the  declination  circle  readings  were 
1-35°  45'  15"  and  40°  16'  38",  and  therefore  h  equaled 
42°  15'  42".  The  value  of  <£  is  42°  16'  47".  The  axis  was 
therefore  1'  5"  too  low.  The  telescope  was  moved  in 
declination  until  the  verniers  read  40°  17'  45"  and 
220°  17'  30",  and  the  axis  adjusted  until  the  thread  was 
again  central  in  the  hole. 

The  hour  circle  readings  were  24*  2m  55*.5  and  23*  57™  I5, 
and  their  mean  was  23*  59m  585.2.  The  error  was  there- 
fore — 1*.8.  The  verniers  were  moved  to  the  west  2s. 

A  repetition  of  the  observations  gave  h  =  42°  16'  49", 
and  the  mean  of  the  hour  circle  readings,  24*  0TO  Os.5. 
Further  adjustment  was  not  required.  The  index  error  of 
the  hour  circle  was  -f-  0*.5,  and  the  index  correction  to  be 
applied  to  future  readings  was  —  05.5. 


147.    To  determine  the  azimuth  correction  of  the  vertical 
plane  containing  the  polar  axis.     This  is  best  determined 


218  PRACTICAL   ASTRONOMY 

by  observations  on  one  of  the  four  Ephemeris  circumpolar 
stars  near  its  culmination. 

Using  the  micrometer  eyepiece  (§  60),  direct  the  tele- 
scope to  the  star  a  few  minutes  before  its  culmination, 
note  the  chronometer  time  6^  when  the  star  is  on  the  point 
of  intersection  of  the  wires  (or  any  well  denned  point  in 
the  eyepiece),  and  read  the  hour  circle.  Reverse  the  tele- 
scope to  the  other  side  of  the  pier,  note  the  time  02  when 
the  star  is  at  the  same  point  of  the  eyepiece,  and  read  the 
hour  circle.  Let  ^  and  t2  be  the  hour  circle  readings  in 
the  two  positions,  corrected  for  index  error,  if  any ;  let  a 
be  the  required  azimuth  correction ;  and  let  A0  be  the 
known  chronometer  correction  [see  §152]. 

Neglecting  the  quantities  which  are  eliminated  by  the 
reversal,  we  have  for  the  sidereal  times  when  the  star  is  in 
the  vertical  plane  of  the  polar  axis, 

$l  +  A0  -  tv 

02   +    A0   -    ty 

Therefore,  as  with  the  transit  instrument,  §  98, 

aA  =a-(0l  +  *0-tjt 
aA  =a-(02  +  A0-*2), 

in  which  A  is  given  by,  (222)  and  (234), ' 

sin  (<£  T  8) 

A  — s ' 

COS  0 

the  lower  sign  being  for  lower  culmination.  Solving  for 
a  we  find 

a  =  [a  -  K*i  +  02)  -  A0  +  |  ft  +  «,)]  B.n°™*8)-          (313> 

a  is  expressed  in  time:  in  arc,  the  azimuth  correction 
is  15  a. 

If  a  is  4-  i  the  south  end  of  the  axis  requires  to  be  moved 
to  the  west;  if  — ,  to  the  east.  This  is  readily  done. 
Direct  the  telescope  to  a  distant  terrestrial  object  nearly 
in  the  horizon,  make  the  movable  micrometer  wire  vertical, 


ADJUSTMENTS  219 

and  set  it  on  the  object.  Next  move  the  wire  through 
the  distance  a  in  the  proper  direction.  This  can  be  done 
when  the  value  of  one  revolution  of  the  screw  is  known 
[§  61].  Shift  the  whole  instrument  in  azimuth  by  the 
proper  screws  until  the  micrometer  wire  is  again  on  the 
object.  The  vertical  plane  of  the  polar  axis  should  now 
coincide  with  the  meridian. 

If  the  value  of  a  is  large  the  observations  should  be 
repeated.  If  a  is  less  than  3s,  it  will  cause  no  inconven- 
ience and  scarcely  need  be  corrected. 

Example.  Wednesday,  1891  February  25.  51  Cephei 
was  observed  at  upper  culmination  with  the  6-inch  equa- 
torial of  the  Detroit  Observatory,  as  below.  Determine 
the  azimuth  correction.  The  value  of  A0  was  -1-  14'"  36s.O. 

Telescope          Hour  circle  Chronometer 

West  23»  56«  31s  6*  31™  42* 

East  24     0    42  6  35    29 

The  index  correction  of  the  hour  circle  was  —  05.5. 

Amer.  Ephem.,  p.  303,  a  6»  49™  27'.5  8  +  87°  13' 

i(0,  +  <92)  6   33    35.5  <f>  +42    17 

A0  +    14    36.0  cos  8  8.6863 

H'i  +  '2)  23   58    36.0  sin(<£-S)  9.8490n 

_8.0      ^1? 0.069 

sin(<£  -  8) 

The  value  of  a  was  --  0.069  x  -  85.0  =  +  05.6  =  +  9"  ; 
that  is,  the  south  end  of  the  axis  should  be  moved  9"  west. 
This  was  too  small  to  require  correction. 

148.  To  adjust  the  declination  verniers.  Direct  the  tele- 
scope to  a  star,  nearly  in  the  zenith,  whose  declination  is 
known.  Bring  the  star  to  the  center  of  the  eyepiece, 
using  a  high  power,  and  clamp  the  instrument  in  declina- 
tion. Set  the  verniers  so  that  they  read  the  star's  declina- 
tion. They  will  then  be  in  adjustment. 


220  PRACTICAL   ASTRONOMY 

149.  To  center  the  object  glass.     Imperfect  images  are 
often  due  to  the  fact  that  the  object  glass  is  not  properly 
centered.     To  test  this  adjustment,  remove  the  eyepiece 
and  hold  a  candle  flame  in  such  a  position  that  the  images 
of  the  flame  reflected  from  the  surfaces  of  the  object  glass 
can  be  seen  through  the  flame.     If  the  object  glass  is  per- 
fectly centered  all  the  images  should  coincide  when  the 
observer's  eye  and  the  center  of  the  flame  are  in  the  axis 
of  the  telescope.     If  they  do  not  coincide,  raise  one  side  of 
the  object  glass  cell  by  the  set  screws  until  the  coincidence 
is  perfect. 

150.  The  magnifying  power  of  a  telescope  is  equal  to  the 
focal  length  of  the  objective  divided  by  the  focal  length 
of  the  eyepiece.     It  is  therefore  different  for  different  eye- 
pieces on  the  same  telescope,  or  for  the  same  eyepiece  on 
different  telescopes.     The  following  method  of  determin- 
ing it  is  simple,  and  abundantly  accurate. 

Focus  the  telescope  on  a  distant  object,  and  direct  it  in 
the  daytime  to  the  bright  sky.  Hold  a  piece  of  thin,  un- 
glazed  paper  in  front  of  the  eyepiece  at  such  a  distance 
that  the  bright  disk  formed  on  it  is  clearly  defined.  This 
disk  is  the  minified  image  of  the  object  glass.  Measure  its 
diameter  by  a  finely  divided  scale  held  against  the  paper, 
and  measure  the  diameter  of  the  object  glass.  It  can  be 
shown  that  these  diameters  are  to  each  other  as  the  focal 
lengths  of  the  eyepiece  and  object  glass.  Their  quotient 
is  therefore  the  magnifying  power.  Thus,  for  the  equa- 
torial mentioned  above,  the  diameter  of  the  object  glass  is 
6.05  inches,  and  the  diameter  of  the  bright  disk  for  a  cer- 
tain eyepiece  is  0.08  inch.  The  magnifying  power  is, 
therefore,  6.05  -f-  0.08  =  76. 

A  definite  statement  of  the  magnifying  power  to  be  used 
in  observing  an  object  cannot  be  made.  A  higher  power 
can  be  used  when  the  seeing  is  good,  i.e.,  when  the  images 
in  the  telescope  are  steady  and  well  defined,  than  when 


CHRONOMETER   CORRECTION  221 

the  seeing  is  poor.  Lower  powers  must  in  general  be  used 
with  nebulse  and  comets.  The  very  highest  powers  can  be 
used  with  stars  and  some  of  the  planetary  nebulse,  if  the 
seeing  is  good.  Further  than  this,  the  observer  must  select 
that  eyepiece  which  on  trial  gives  the  best  results. 

151.  The  field  of  view  is  the  circular  portion  of  the  sky 
which  can  be  seen  through  the  telescope  at  one  time.     Its 
diameter  is  equal  to  the  angle  contained  by  two  rays  drawn 
from  the  center  of  the  object  glass  to  the  two  extremities 
of  a  diameter  of  the  eyepiece.     The  diameter,  expressed  in 
arc,  is  equal  to  15  times  the  interval  of  time  required  by 
an  equatorial  star  to  traverse  it.     This  can  be  directly 
observed. 

152.  The  chronometer  correction  is  quickly  obtained,  with 
an  accuracy  sufficient  for  all  ordinary  uses  of  the  equa- 
torial, by  the  following  method  : 

Direct  the  telescope  to  a  known  star  nearly  in  the 
zenith,  note  the  chronometer  time  0l  when  the  star  is  on 
the  point  of  intersection  of  the  wires,  and  read  the  hour 
circle.  Carry  the  telescope  to  the  other  side  of  the  pier, 
observe  the  star  as  before  at  the  time  #2,  and  read  the  hour 
circle.  Let  the  hour  circle  readings  corrected  for  index 
error  be  ^  and  £2.  We  have,  by  (40), 

a  =  0,  +  A0  -  tr 
a  =  0,  +  A<9  -  *2, 

neglecting  only  very  small  quantities  and  those  eliminated 
by  reversal.  Therefore 


For  many  purposes  an  observation  on  one  side  of  the 
pier  will  suffice,  and  we  have 

A0  =  a  +  *!-#!.  (315) 

Example.   Ann  Arbor,  Wednesday,  1891  Feb.  25.     The 
following  observation  of  Castor  was  made  with  the  6-inch 


222  PRACTICAL  ASTRONOMY 

equatorial,  to  determine  an  approximate  value  of  the  chro- 
nometer correction. 

Chronometer  time,       Ol    7     5m  9« 
Hour  circle,  ^  23   52    3 

Amer.  Ephem.,  p.  327,  a     7  27  39 

Therefore,  by  (315),  A0  =  +  14™  33s. 

153.  To  direct  the  telescope  to  an  object  whose  right 
ascension  (a)  and  declination  (S)  are  known,  first  deter- 
mine whether  the  object  is  east  or  west  of  the  meridian. 
If  the  right  ascension  is  greater  than  the  sidereal  time,  it 
is  east ;  if  less,  it  is  west.  Generally,  if  the  object  is  east, 
the  telescope  should  be  west  of  the  pier ;  if  the  object  is 
west,  the  telescope  should  be  east  of  the  pier.  Move  the 
telescope  in  declination  till  the  declination  circle  reads  B. 
To  the  reading  of  the  chronometer  add  the  chronometer 
correction,  and  one  or  two  minutes  more  for  the  time  con- 
sumed in  setting.  Subtract  a  from  this  sum  and  set  off 
the  difference  (which  is  the  hour  angle)  on  the  hour  circle. 
When  the  chronometer  indicates  the  time  for  which  the 
hour  angle  was  computed,  the  object  should  be  seen  in  the 
finder.  Move  the  telescope  until  the  star  is  on  the  inter- 
section of  the  finder  cross-wires.  The  star  should  then  be 
visible  near  the  center  of  the  field  of  view  of  the  (prin- 
cipal) telescope. 

Conversely,  if  an  unknown  star  is  seen  in  the  telescope, 
the  chronometer  time  noted  and  the  circle  readings  taken : 
then  the  declination  circle  reading  is  the  star's  declination  ; 
and  the  chronometer  time  of  observation,  plus  the  chro- 
nometer correction,  minus  the  hour  circle  reading,  is  its 
right  ascension. 

These  results  are  only  approximate,  of  course,  since  the 
instrument  will  never  be  in  perfect  adjustment,  and  the 
star  will  not  be  seen  in  its  true  place,  owing  to  the  refrac- 
tion, etc. 


DETERMINATION   OF   APPARENT   PLACE  223 


DETERMINATION   OF   APPARENT   PLACE    OF   AN   OBJECT 

154.  By  the  method  of  micrometer  transits.  Select  a 
known  *  star,  called  a  comparison  star,  whose  right  ascen- 
sion and  especially  whose  declination  differ  .as  little  as 
possible  from  that  of  the  object.  Revolve  the  filar  microm- 
eter [§  60]  until  the  star  in  its  diurnal  motion  follows 
along  the  micrometer  wire.  The  wire  in  this  position  is 
exactly  east  and  west,  or  parallel  to  the  equator,  and  the 
reading  of  the  position  circle  for  this  direction  of  the  wires 
is  called  the  equator  reading.  If  the  object  and  comparison 
star  are  in  the  vicinity  of  the  pole,  their  diurnal  circles 
will  be  sharply  curved,  and  in  this  case  the  equator  read- 
ing of  the  circle  should,  first  of  all,  be  determined  from  an 
equatorial  star.  Direct  the  telescope  just  in  advance  of 
the  two  objects.  The  diurnal  motion  will  carry  them  across 
the  field.  Note  the  chronometer  times  when  they  cross  the 
transverse  wire  or  wires.  The  difference  of  these  times 
for  the  two  objects  is  the  difference  of  their  right  ascen- 
sions. Also,  when  the  first  or  preceding  object  approaches 
the  center  of  the  field,  move  the  whole  system  of  wires 
until  the  object  follows  along  the  fixed  wire.  When  the 
second  or  following  object  approaches  the  center  of  the 
field,  bisect  it  with  the  micrometer  wire.  Read  the  microm- 
eter in  this  position,  and  also  when  the  micrometer  wire 
is  in  coincidence  with  the  fixed  wire.  The  difference  of 
the  two  readings,  multiplied  by  the  value  of  a  revolution 
of  the  screw  [§  61],  is  the  difference  of  the  declinations  of 
the  two  objects. 

Care  should  be  taken  to  have  the  micrometer  and  fixed 
wires  exactly  parallel, j-  and  the  transverse  wire  (or  wires) 
exactly  perpendicular  to  the  micrometer  wire.  To  test  the 

*  That  is,  a  star  whose  accurate  position  is  given  in  one  or  more  of  the 
star  catalogues. 

t  In  good  forms  of  the  micrometer,  an  adjusting  screw  is  provided  for 
bringing  them  into  parallelism. 


224  PBACTICAL   ASTRONOMY 

relative  positions  of  the  two  sets  of  wires,  direct  the  tele- 
scope to  an  equatorial  star,  adjust  the  micrometer  wire  so 
that  the  star's  diurnal  motion  will  carry  it  across  the  field 
in  coincidence  with  the  wire,  and  read  both  verniers  of  the 
position  circle.  Rotate  the  system  of  wires,  adjust  the 
transverse  wire  so  that  the  star  will  cross  the  field  in  coin- 
cidence with  the  wire,  and  read  both  verniers.  The  dif- 
ference of  the  two  position  circle  readings  should  be 
exactly  90°. 

Many  observers  prefer  to  observe  only  one  coordinate  at 
a  time.  A  good  program  to  follow  is,  measure  the  differ- 
ence of  declination,  revolve  the  micrometer  90°  and  ob- 
serve the  difference  of  right  ascension,  then  revolve  90a 
more  and  measure  the  difference  of  declination  again. 

In  any  case,  the  observations  should  be  repeated  several 
times,  and  the  mean  of  all  the  observations  adopted.  If 
the  object  has  a  proper  motion,  the  differences  in  right 
ascension  and  declination  are  those  corresponding  to  the 
instant  when  that  object  was  observed :  that  is,  the  mean  of 
the  chronometer  times  for  the  object,  plus  the  chronometer 
correction. 

The  mean  place  of  the  comparison  star  *  will  be  given 
for  the  epoch  of  the  catalogue  which  contains  it.  Reduc- 
ing this  to  the  mean  place  for  the  beginning  of  the  year 
of  observation  by  §  46,  47  or  52,  thence  to  the  apparent 
place  for  the  instant  of  observation  by  §  55,  and  applying 
the  micrometer  differences  to  the  apparent  place,  we  obtain 
the  observed  place  of  the  object.  This  must  be  corrected 
for  refraction  and  parallax. 

The  refraction  correction  will  be  small,  since  the  star 


*  If  the  star  is  a  very  bright  one,  it  may  be  identified  satisfactorily, 
both  in  the  sky  and  in  the  catalogue,  by  the  methods  of  §  153.  But,  in 
case  it  is  faint,  the  observer  should  always  compare  a  chart  of  the  neigh- 
boring stars,  prepared  from  the  catalogue,  with  that  region  of  the  sky, 
making  sure  that  the  configurations  of  the  stars  on  the  chart  and  in  the 
sky  agree. 


DETERMINATION    OF    APPARENT   PLACE 


225 


and  object  will  be  refracted  nearly  the  same  amount  in 
nearly  the  same  direction.  An  equatorial  is  a  fixed  part 
of  an  observatory,  and  tables  of  differential  refractions  in 
right  ascension  and  declination  in  every  part  of  the  sky 
should  be  computed  for  the  latitude  of  the  observatory. 
The  corrections  can  then  be  taken  from  the  tables  very 
quickly. 

Until  such  tables  are  constructed,  the  corrections  can 
be  computed  by  the  following  method:  Let  tQ  be  the 
mean  of  the  hour  angles  of  the  star  and  object,  BQ  the  mean 
of  their  declinations,  and  z  the  mean  of  their  zenith  dis- 
tances. Substitute  these  values  of  t  and  B  in  (35),  (36) 
and  (37),  to  determine  L  and  z.  The  corrections  to  the 
observed  place  will  be  given  by  * 

A     -A  B'  -B  tan  t0  sin  L  cos  (2  80  +  L) 

a~15'     sin2  ($n  +  L)'    ~ 


cos'2  80 


(316) 


-8 


(317) 


in  which  B'  —  B  is  the  declination  of  the  object  minus  the 
declination  of  the  star  expressed  in  seconds  of  arc,  and  K 

is  defined  by 

(318) 


.#,  T  and  7  have  the  same  significance  as  in  §  30,  and  their 
values  are  given  in  TABLE  I  of  the  Appendix.  The  values 
of  log  p"  and  X"  are  tabulated  below  with  the  argument  z. 


z 

log  /*" 

X" 

z 

log  n" 

X" 

0° 

6.446 

1.00 

80° 

6.395 

1.10 

45 

6.444 

1.00 

82 

6.370 

1.15 

60 

6.440 

1.01 

83 

6.351 

1.18 

70 

6.433 

1.03 

84 

6.323 

1.21 

75 

6.422 

1.05 

85 

6.285 

1.24 

*  These  equations  are  derived  in  Chauvenet"1 s  Spherical  and  Practical 
Astronomy,  Vol.  II,  pp.  450-460. 
Q 


226  PRACTICAL   ASTRONOMY 

It  is  only  in  the  most  refined  measurements  and  in 
extreme  states  of  the  weather  that  the  barometer  and 
thermometer  readings  need  be  taken  into  account.  With 
comets  it  will  scarcely  ever  be  necessary,  except  when 
they  are  very  near  the  horizon. 

If  one  or  both  of  the  bodies  is  in  the  solar  system,  and 
at  different  distances  from  the  observer,  the  observations 
will  require  correction .  for  parallax,  by  the  methods  ex- 
plained in  full  in  §  28. 

Four-place  tables  are  sufficient  for  computing  the  refrac- 
tion and  parallax  corrections. 

Example.  Wednesday,  1890  July  23,  the  author  made 
the  following  observation  of  Coggia's  comet  with  the 
12-inch  equatorial  of  the  Lick  Observatory.  Required  its 
apparent  place. 

The  comet  was  south  of  and  preceding  the  6th  magni- 
tude star  No.  1518  Pulkowa  Catalogue  for  1855.0.  The 
reading  of  the  position  circle  when  the  star  followed 
along  the  micrometer  wire  was  201°. 35.  The  micrometer 
readings  when  the  micrometer  and  fixed  wires  coincided 

were 

19.947 

.947 

.947 

Mean  19.947 

When  the  comet  was  in  the  center  of  the  field  the  fixed 
wire  was  made  to  bisect  it,  and  the  chronometer  time  was 
noted.  When  the  star  reached  the  center  of  the  field 
(nearly  four  minutes  later)  the  micrometer  wire  was  made 
to  bisect  it,  and  the  micrometer  reading  noted.  In  this 
way  the  difference  of  the  declinations  was  observed,  as 

below. 

Chronometer  Micrometer  Remarks 

16»  43"*  11-                  22.954  Very  windy 

48    59                   23.822  Very  windy 

54      2                   24.550  Very  windy 
Means    16  48    44                  23.775 


DETERMINATION  OF  APPARENT  PLACE 


227 


The  micrometer  was  rotated  90°  until  the  circle  read 
291°. 35,  and  the  chronometer  times  of  transit  over  the  two 
wires  noted,  as  below. 


Comet 

Star 

Difference 

17*    0»5K 

7 

Qm 

59*.2 

17*    4m 

44*.6 

4* 

'  51-.8 

-3* 

52'.75 

5 

42. 

3 

5 

49 

.5 

9 

34.0 

9 

41.3 

3 

51.75 

10 

37. 

8* 

10 

40 

.1 

14 

28.0 

14 

36.3 

3 

50  .20 

23 

3. 

5 

23 

11 

.5 

26 

49.8 

26 

58.1 

3 

46.45 

27 

39. 

5* 

27 

52 

.2 

31 

24.6 

31 

37.2 

-3 

45.05 

Means 

17*  13*  39* 

-3 

49.24 

The  micrometer  was  rotated  90°  further  until  it  read 
21°. 35,  and  the  difference  of  the  declinations  measured 
again,  as  below, 

Micrometer  Remarks 

9.751  Very  windy 

8.867  Very  windy 

7.545  Very  windy 
8.721 

Readings  for  coincidence  of  wires, 

19.944 

.946 

.943 

Mean     19.944 

The  value  of  one  revolution  of  the  screw  is  .S  =  14".058. 
We  shall  combine  the  two  differences  of  declination,  thus : 


Chronometer 

17*  34™  22« 

39    44 

48    40 

Means     17  40    55 


Chronometer 
16*  48™  44« 
17  40    55 
Means     17   14    49 


Diff .  of  decl. 

-  3.828/2 

-  11.223  R 

-  7.525  R  =  -l'45".8. 


The  chronometer  time  for  the  declinations  is  lw  105  greater 
than  that  for  the  right  ascensions.  From  the  two  measured 
declinations  it  is  found  that  the  declination  changed  2". 3 
in  lm  105.  Therefore,  at  17*  13W  39*  the  difference  of  decli- 
nation was  —  1'  43".5. 


*  The  distance  between  the  wires  was  changed  intentionally. 


228  PRACTICAL   ASTRONOMY 

The  mean  place  and  proper  motion  of  the  comparison 
star  for  1855.0  given  by  the  catalogue  were 

a  =  9"  29'"  17*.57,         8  =  +  40°  53'  16  '.1, 
p  =  -  (K0022,  fji'  =  +  0".008. 

The  mean  place  for  1890.0  was,  by  §  47, 

a  =  9*  31"  29".56,         8  =  +  40°  43'  58".8  ; 

and  the  apparent  place  for  sidereal  time  1890  July  23d  17/J, 

by  §55, 

a'  =  9A  31™  28«.90,        8'  =  +  40°  44'  4".2. 

Therefore  the  observed  place  of  the  comet  at  17A  13m  395  was 

a  =  9*  27™  39'.66,         8  =  +  40°  42'  20".7. 
The  chronometer  correction  was  —  1"'  19s.     We  have 

Chronometer  time,     ff  17*  13»  39s 

Chronometer  corr.,  A0  1    19 

Sidereal  time,              0  17   12    20 

Right  ascension,          a  9   27    40 

Hour  angle,                  t  7   44    40 

The  corrections  for  differential  refraction  corresponding 
to  this  hour  angle  and  declination,  taken  from  the  tables 
constructed  for  the  Lick  Observatory,  or  computed  from 
(316)  and  (317),  are 

Aa  =  -  (K04,        AS  =  -  0".6. 

The  corrections  for  parallax  at  the  unit  distance,  i.e.  the 
parallax  factors,  taken  from  tables  constructed  for  the  Lick 
Observatory,  or  computed  from  (92),  (90)  and  (93),  are 

Aa  =  +  Os.56,         AS  =  +  6".l. 

The  comet's  distance  was  1.57,  and  therefore  the  required 
corrections  for  parallax  are 

Aa=  +  0".36,         AS  =  +  3  .9, 


DETERMINATION  OF  APPARENT  PLACE      229 

Applying  these  corrections  to  the  observed  place  we  ob- 
tain the  following  apparent  place  of  the  comet : 

Mt.  Hamilton  sid.  time  Apparent  a  Apparent  8 

1890  July  23,  17*  13™  39'  9*  27m  39'.98          -f  40°  42'  24".0 

155.  By  the  method  of  direct  micrometer  measurement. 
When  the  object  whose  position,  (a",  S"),  is  to  be  deter- 
mined is  comparatively  near  the  comparison  star,  (a',  £'), 
so  that  both  are  well  within  the  field  of  view,  their  differ- 
ences of  right  ascension  and  declination  are  conveniently 
determined  by  direct  micrometer  measurement. 

Let  the  micrometer  wire  be  placed  parallel  to  the  equa- 
tor,, as  before.  By  means  of  the  driving  clock  keep  the 
telescope  directed  to  the  star  and  object  so  that  the  point 
midway  between  them  will  be  as  nearly  as  possible  in  the 
center  of  the  field.  Bisect  the  star's  image  with  the  fixed 
wire  and  the  object's  image  with  the  micrometer  wire,  and 
note  the  chronometer  time  and  the  reading  m  of  the  microm- 
eter. If  mQ  is  the  reading  for  coincidence  of  the  wires, 
and  R  the  value  of  a  revolution  of  the  screw,  then  the 
apparent  difference  AS  of  the  declinations  is  given  by 

A8  =  (m  -  m^R.  (319) 

Rotate  the  wires  through  90°,  bisect  the  two  images  as 
before  and  note  the  time  and  the  micrometer  reading  m' . 
The  apparent  difference  of  the  right  ascensions  is  given  by 

Aa  =  (m>  -  m0)  R  sec  f  (S",+  8')-  (320) 

This  method  cannot  be  used  with  safety  near  the  pole 
unless  the  instrument  is  in  good  adjustment  and  the  differ- 
ence of  right  ascension  is  small. 

The  apparent  differences  of  right  ascension  and  declina- 
tion will  require  correction  for  differential  refraction.  The 
corrections  could  be  computed  by  differential  formulae,  but 
an  equally  satisfactory  method  consists  in  computing  the 
absolute  refractions  4n  right  ascension  and  declination  for 
the  star  and  object,  and  taking  their  differences  as  the  cor- 


230  PRACTICAL    ASTRONOMY 

rections  to  the  observed  intervals  Aa  and  AS.  The  values 
of  the  parallactic  angles  and  zenith  distances,  q',  z'  for 
the  star  and  q",  z"  for  the  object,  may  be  taken  from 
general  tables  constructed  for  the  point  of  observation,  or 
computed  by  (35),  (36)  and  (37).  These  should  be  par- 
tially checked  by  the  formula 

z"  -  z'  =  Az  =  -  cos  8'  sin  q'  Act  -  cos  q'  AS,  (321) 

formed  by  differentiating  (30)  and  (41)  and  combining 
the  results  with  (31).  The  refraction  r'  for  the  star  may 
be  computed  by  means  of  (95),  (96)  or  (97),  remembering 
that  z  in  these  formulae  is  the  apparent  zenith  distance. 
Formula  (96)  will  be  sufficiently  accurate,  except  in  case 
of  observations  made  very  near  the  horizon.  The  refrac- 
tion r"  for  the  object  should  now  be  found  differentially. 
The  change  Ar  in  refraction  due  to  a  change  Az  in  zenith 
distance  is  obtained  from  (96),  thus : 

Ar  =   988&    sec2z  sin  1"  Az,  (322) 

4oO  -(-  t 

in  which  Ar  and  Az  are  expressed  in  terms  of  the  same 
unit.  The  refraction  for  the  object  will  be  given  by 

r"  =  r'  +  Ar.  (323) 

The  corrections  to  the  apparent  places  may  be  obtained 
from  (100)  and  (101),  thus : 

da.'  =-rf  sin  q'  sec  8',  d&  =  -  r1  cos  q1,  (324) 

da"  =  -  r"  sin  q"  sec  8",         d8"  =  -  r"  cos  q".  (325) 

Therefore,  we  shall  have  the  true  differences  of  right  ascen- 
sion and  declination,  as  seen  from  the  point  of  observation, 

a"  -  a'  =  Aa  +  (da"  -  da'),  (326) 

8"  -  8'  =  AS  +  (dS"  -  </8')»  (327) 

from  which  the  values  of  a"  and  B"  may  be  obtained. 

If  the  object  is  in  the  solar  system,  it  will  further  require 
correction  for  parallax. 


DETERMINATION    OF    APPARENT   PLACE  281 

Example.  Lick  Observatory,  Friday,  1898  Nov.  11. 
12-inch  equatorial.  Observer,  W.  J.  Hussey.  The  differ- 
ences of  right  ascension  and  declination  of  the  minor  planet 
Eros  and  the  star  DM.  —  4°.5413  were  measured  directly 
with  the  micrometer,  to  determine  the  apparent  place  of 
the  planet. 

The  mean  of  five  measures  of  difference  of  declination 
was,  (the  planet  being  north  of  the  star), 

m  =  55r.007  at  (sidereal)  chronometer  time  22*  4Sm  21*.0. 
The  mean  of  ten  measures  of  difference  of  right  ascension 
was,  (the  planet  being  west  of  the  star), 

m'  =  67»-.lSO  at  chronometer  time  22*  52ro22«.7. 
The   mean  of   five  additional   measures  of   difference  of 
declination  was 

m  =  55»-.228  at  chronometer  time  22*  56™278.2. 
The  coincidence  of  the  wires  was  at  46r.650  ;  the  value  of 
one  revolution  of  the  screw  is  14".  058  ;  the  chronometer 
correction  was  -f-  2m  53S.7  ;  and  the  apparent  place  of  the 
star  for  the  instant  of  observation  was  [from  Karlsruhe 
Beobachtungeri]  , 

a'  =  21*  13™  10-.63,  8'  =  -  4°  6'  10".9. 

The  observations  for  determining  AS  will  be  combined 
as  follows,  the  reductions  —  I5.  4  and  —  Or.001  being  applied 
so  that  the  declination  observations  will  refer  to  the  same 
instant  as  the  right  ascension  observations. 

Chronometer,  22*  48™  2KO         m,  55'.007 

22  56  27.2  55.228 

Means,  22  52  24.1  55  .118 

Reductions,  -1.4  -  .001 

22  52  22.7  55.117 

M  46  .650 


A8,  +  8  .467  R  =  +  119".04 
Chronometer,  22  52   22  .7       m',      67  .180 
Chron.  corr.,        +2    53  .7        m0,     46  .650 
Sid.  time,  0,     22  55   16.4       Aa,  -  20  .530  R  sec  (  -  4°  5  .2) 

Aa,  -  289".36 


232  PRACTICAL   ASTRONOMY 

Applying  these  values  of  A«  and  AS  to  the  star's  place  we 
have  the  approximate  position  of  the  planet, 

a"  =  2P  12™  51«.3,  8"  =  -  4°  4;  12". 

The  data  for  solving  (35),  (36)  and  (37)  are 

<j>  =      37°  20'  26"  0  =      22*  55W  IQ'A 

t'  =+    1*42™  6«  t"  =  +    1    42   25 

t'  =  +  25°  31'  30"  t"  =  4-  25°  36'  15" 

8'  =  -   4    6  11  8"  =  -   4     4  12 

The  quantities  obtained  from  the  solution  are 

q'  =  27°  33'  50"  q"  =  27°  38'  46'' 

zf  =  47  45  42  z"  =  47  46  10 

The  value  of  z  "  —  z'  =  Az  =  +  28"   agrees  exactly  with 
that  obtained  from  (321). 

The  mean  value  of  the  refraction  at  true  zenith  distance 
z'  =  47°  46'  is  about  1'.  Solving  (96),  (322)  and  (323) 
for  z  =  47°  45'  and  Az  =  +  28",  assuming  b  =  25.8  inches 
and  t  =  +  55°  as  the  average  for  observing  weather  at 
that  season  of  the  year,  we  obtain 

r'  =  54".2, 

r"  =  54  .2  +  0".015  =  54".215. 

Substituting  these  in  (324)  and  (325)  we  obtain 

da'  =  -  25".14,        d&  =  -  48".05, 
da"  =  -25  .22,         d8»  =  -  48  .03 ; 

and,  therefore,  from  (326)  and  (327), 

aii  -a'  =  -  289".36  -  0".08  =  -  289".44  =  -  19'.30, 
8'*  _  8'  =  +  119  .04  -}-  0  .02  =  +  119  .06  =  +    1'  59".l ; 
whence 

a"  =  21*  12™  51-.33,        B  =  -  4°  4'  11".8. 

The  distance  A  of  the  planet  from  the  earth  being  1.225 
units,  the  parallax  corrections  taken  from  general  tables 
or  computed  from  (89),  (90)  and  (91),  are  +  Os.17  and 


POSITION   ANGLE    AND   DISTANCE 

-f  4". 7.     The  apparent  place  of  the  planet  referred  to  the 
center  of  the  earth  was  therefore 

Mt.  Hamilton  sidereal  time          Apparent  a          Apparent  8 
1898  Nov.  11,  22*  55"*  16'.4         21*  12™  51'.50        -  4°  4'  7 ".I 


DETERMINATION    OF   POSITION    ANGLE   AND    DISTANCE 

156.  The  relative  positions  of  two  objects  close  together 
are  conveniently  expressed  in  terms  of  position  angle  and 
distance.  The  position  angle,  jt?,  of  one  star,  B,  with  refer- 
ence to  another  star,  A,  is  the  angle  which  the  great  circle 
passing  through  the  two  objects  makes  with  the  hour 
circle  passing  through  A,  reckoned  from  the  north  toward 
the  east  through  360°.  Their  distance,  s,  is  the  length  of 
the  arc  of  the  great  circle  joining  them. 

To  determine  the  position  angle,  revolve  the  micrometer 
until  one  of  the  stars,  by  its  diurnal  motion,  follows  along 
the  micrometer  wire,  and  note  the  reading  P0  of  the  posi- 
tion circle.  Keeping  the  telescope  directed  upon  the  stars 
by  means  of  the  driving  clock,  revolve  the  micrometer 
until  the  micrometer  wire  passes  through  the  two  stars* 
and  note  the  circle  reading  P.  The  position  angle  re- 
quired is 

jt>  =  P-(P0±90°).  (328) 

To  determine  the  distance,  revolve  the  micrometer  to 
the  circle  reading  P  ±  90°.  Bisect  one  of  the  stars  with 
the  fixed  wire  by  moving  the  whole  system  of  wires,  then 
bisect  the  other  star  with  the  micrometer  wire,  and  note 
the  reading  m  of  the  micrometer.  If  ra0  is  the  reading  of 
the  micrometer  when  the  two  wires  coincide,  and  R  the 
value  of  a  revolution  of  the  screw,  the  required  distance  is 

s  -  (m  -  w?0)  R.  (329) 

In  very  accurate  measurements  bisect  the  two  stars  as 
above,  and  take  the  reading  m.  Move  the  micrometer 
wire  to  the  other  side  of  the  fixed  wire,  bisect  the  stars 


234  PRACTICAL    ASTRONOMY 

witli  the  wires  in  that  order,  and  take  the  reading  m'. 
The  distance  is  now  given  by 

s  =  $(m'-m)R.  (330) 

111  this  way  several  systematic  and  personal  errors  are 
eliminated,  partially  at  least.  This  method  is  called  the 
method  of  double  distances. 

The  values  of  s  and  p  will  be  affected  by  refraction ; 
but  in  the  case  of  double  stars,  to  which  the  method  is 
especially  applied,  the  correction  for  refraction  may  usually 
be  neglected. 

If  the  distance  between  the  two  stars  is  large,  the  tele- 
scope should  be  directed  so  that  the  two  stars  will  fall  on 
opposite  sides  of  the  center  of  the  field,  and  at  equal  dis- 
tances from  the  center.  In  this  case  the  measured  position 
angle  is  the  angle  between  the  arc  joining  the  two  stars, 
and  the  hour  circle  passing  through  the  middle  point  of 
that  arc.  Let  p'  and  s'  be  the  observed  angle  and  dis- 
tance. Let  their  values  correcfed  for  refraction  be  p  and 
8.  Let  z  and  q  be  determined  for  the  point  midwa}^  be- 
tween the  two  stars  from  (35),  (36)  and  (37),  and  let  K  be 
defined  as  in  (318).  It  can  be  shown  *  that 

p  =  p'  —  K  cosec  1"  [tan2  z  cos(p'  —  q}  sin  (p1  —  9) 

+  tan  z  sin  q  tan  £  (8  +  S')l»  (331) 

s  =  s1  +  SK  [tan2  z  cos2  (p1  -  q)  +  1] .  (332) 

This  value  of  p,  referring  to  the  point  midway  between 
the  two  stars,  may  readily  be  converted  into  the  position 
angle  of  each  star  with  reference  to  the  other  star.  Let 
S',  in  the  position  a',  Sr,  represent  the  western  star;  S", 
in  the  position  a",  8rr,  the  eastern  star ;  M  the  point  mid- 
way between  them ;  and  P  the  pole  of  the  equator.  Let 
p1  be  the  position  angle  of  the  eastern  star  with  reference 
to  the  western,  and  180°  +  p"  the  position  angle  of  the 

*  See  Chauvenet's  Sph.  and  Prac.  Astronomy,  Vol.  II,  pp.  450-459. 


POSITION   ANGLE   AND   DISTANCE  235 

western  star  with  reference  to  the  eastern.  In  practice, 
the  declination  of  one  star  will  always  be  known ;  and  if 
the  declination  of  the  other  is  unknown,  its  value  may  be 
found  with  sufficient  accuracy  from  equation  (338)  below. 
Let  SQ  =  J(&"  +  &')•  Without  sensible  error,  we  may 
assume  80  as  the  declination  of  the  point  M  defined  above. 
Then,  from  the  triangles  PS'M  and  PS"M  we  can  write 
[Chauvenet's  Sph.  Trig.,  §  70,  (N)], 

tan^'  =  cos  ^  cos PT  sin  i.  tan  8;  (333) 


— .    l    f      -,  (334) 

cos  %s  cosp  —  sin  \  s  tan  d0 

which  determine  pr  and  plf.  Their  values  will  differ  very 
little  from  p,  unless  the  stars  are  very  near  the  pole. 

It  is  frequently  required  to  convert  position  angle  and 
distance  into  the  corresponding  differences  of  right  ascen- 
sion and  declination.  From  the  triangle  PS'iS"  defined 
above,  we  may  write  [Chauvenetfs  8ph.  Trig.,  (44)], 

sin  I  (a"  -  a')  =  sin  %  s  sin  \  (p"  +  p')  sec  80,  (335) 

sin  i  (8"  -  8')  =  sin  i  5  cos  f  (/>"  +  p'")  sec  £  (a"  -  a'),      (336) 

which  solve  the  problem.  If  the  stars  are  at  some  distance 
from  the  pole,  we  may  safely  substitute  p  for  ^(pfr  +  p') 
in  these  equations. 

If  the  stars  are  not  far  from  the  equator,  or  if  s  is  rela- 
tively small,  or  if  only  moderate  accuracy  is  required, 
these  equations  may  be  written, 

a"  —a!  =  samp  sec  80,  (337) 

8"-  8'  =scos^.  (338) 

Example.  36-inch  equatorial,  Lick  Observatory,  Thurs- 
day night,  1898  November  17.  Observer,  W.  J.  Hussey. 
The  position  angle  and  distance  of  the  faint  companion 
of  Sirius  with  reference  to  the  principal  component  were 
measured  as  below.  The  distance  was  determined  by  the 


236  PRACTICAL    ASTRONOMY 

method  of  double  distances.  The  equator  reading  of  the 
position  circle  was  111°.  7,  and  the  value  of  a  revolution 
of  the  micrometer  screw  is  9".  907. 


P,  185°.6 

TO',  56'.  199                         w, 

55»-.344 

184.7 

.182 

.350 

184.3 

.180 

.336 

184.7 

.178 

.342 

182  .2 

.193 

.336 

185.0 

Mean  TO',  56  .186             Mean  TO, 

55  .342 

184.3 

w,  55  .342 

MeanP,   184.4 

£  (mr  -  TO),    0'  .422 

P0,  111  .7 

R,    9".907 

72.7 

s,    4".18 

+  90.0 

p,  162°.7 

The  correction 

for  refraction  is  not  appreciable. 

THE   RING  MICROMETER 

157.  This  consists  of  a  narrow  metal  ring,  one  or  both 
of  its  edges  turned  exactly  circular,  attached  to  a  thin 
piece  of  glass  in  the  focal  plane  of  an  eyepiece.     When 
the  eyepiece  is  put  on  the  telescope  and  focused,  the  ring 
is  also  in  the  focal  plane  of  the  object  glass. 

If  the  times  of  transit  of  two  stars  over  the  edges  of  the 
ring  are  observed,  the  differences  of  their  right  ascensions 
and  declinations  can  be  found.  But  results  obtained  in 
this  way  can  be  regarded  as  only  approximately  correct, 
and  the  ring  micrometer  should  never  be  used  with  an 
equatorial  telescope  unless,  in  case  of  great  haste,  there  is 
not  time  to  attach  the  filar  micrometer  and  adjust  its  wires 
by  the  diurnal  motion.  The  principal  advantage  of  the 
ring  micrometer  is  that  it  can  be  used  with  an  instrument 
mounted  in  altitude  and  azimuth  as  well  as  with  an  equa- 
torial, whereas  a  filar  micrometer  cannot. 

158.  To  find  the  radius  of  the  ring.     Select  two  stars 
whose  declinations  are  accurately  known,  the   difference 


RING    MICROMETER  237 

of  whose  declinations  is  a  little  less  than  the  diameter  of 
the  ring,  and  whose  right  ascensions  do  not  differ  more 
than  three  or  four  minutes.  Two  stars  to  fulfil  these  con- 
ditions can  always  be  found  in  the  Pleiades.  When  these 
stars  are  nearly  on  the  observer's  meridian,  observe  their 
transits  over  the  edge  of  the  ring. 

In  Fig.  28  let  ODD'  C'  represent  the  ring;   CD  the  path 
of  one  star  (a,  S),  and  fx  and  t2  the  observed  sidereal  times 
of  its  transit  over  0  and  D;   0'Df  the 
path  of  the  other  star  (a',  £')?  and  tj 
and  t2r  the  times  of  its  transit  over 
C'  and  Dr.     Draw  MM'  perpendicu- 
lar to  CD  and  C'D'.     Draw  the  radii 
CO  and  C'  0,  and  let  r  represent  their 
value  in  seconds  of  arc.     If  we  put 

COM  =  y,        C'OM*  =  y', 

FIG.  28 

we  can  write 

OM  =  r  cos  y,  CM  =  r  sin  y, 

OM'  =  r  cos  y',          C'M'  =  r  sin  y'  ; 

and  therefore 

MM'  =  r  (cos  y'  +  cos  y)  =  2  r  cos  f  (y'  -I-  y)  cos  %  (y'  -  y)  ,  (339) 

C'M'  +  CM  -  r  (sin  y'  +  sin  y)  =  2  r  sin  \  (y'  +  y)  cos  £  (y'  -  y),  (340) 
C'M1-  CM  =  r(siny'-  siny)  =  2r  cos*  (y'-f  y)  sin  *  (y'-  y).  (341) 

We  have 

MM  '  =  8'  -  8,  CM  =  *£-  (t2  -  «!>  cos  8,  CW  =  ^  (V  -  */)  cos  8'  ; 

and  if  we  put 

K/  +  y)  =  ^, 
we  can  write 


tan  A  =  .        =  -i  -iy      (340) 

MM1  8'  —  8 


tan  B  =  ~         =         2        l  lv wa  v       f  343) 

MM1  8'  —  8 

*-*  (344) 


2  cos  A  cos  5     2  cos  A  cos  B 


238  PRACTICAL   ASTRONOMY 

The  apparent  distance  between  the  stars  is  affected  by 
refraction.  Since  the  observations  are  made  near  the 
meridian,  the  refraction  in  right  ascension  can  be  neg- 
lected, and  it  will  be  sufficiently  accurate  to  consider  that 
its  effect  upon  the  difference  of  the  declinations  is  equal 
to  the  difference  of  refraction  in  zenith  distance  of  the 
stars  when  they  are  on  the  meridian.  The  difference  of 
the  declinations  furnished  by  the  star  catalogues  requires 
to  be  decreased  numerically  by  the  difference  of  the  re- 
fractions before  substituting  in  the  above  equations.  The 
difference  6f  the  refractions  may  be  readily  obtained  with 
the  assistance  of  the  table  in  §  111,  based  upon  equation 
(280),  or  from  the  table  of  mean  refractions,  Appendix, 
TABLE  II. 

Example.  Friday,  1889  Jan.  25.  The  following  times 
of  transit  of  23  Tauri  and  27  Tauri  over  the  outer  edge  of 
the  ring  micrometer  of  the  12^-inch  equatorial  of  the 
Detroit  Observatory  were  noted,  to  determine  the  radius 
of  the  ring. 

23  Tauri  27  Tauri 

^  =  3*  30™  41«.5,  V  =  3*  33»  33'.5, 

*2  =  3   30    53  .0,  t2f  =  3  33    41 .0. 

The  mean  places  of  these  stars  for  1850.0  are  given  in 
NewcornVs  Standard  Stars.  Reducing  them  to  the  mean 
place  for  1889.0  by  §  52,  and  thence  to  the  apparent  place 
at  the  instant  of  observation  by  §  55,  (6),  we  obtain 

8  =  23°  36'  4". 13,  8'  =  23°  42'  45".40. 

The  zenith  distance  of  the  stars  is  about  18°  and  the 
difference  of  their  zenith  distances  is  about  7'.  Entering 
the  table  in  §  111  with  1  (zf  -  z")  =  3'.5  and  z  =  18°,  one- 
half  the  difference  of  the  refractions  is  0".07,  or  the  whole 
difference  is  0".14.  Therefore  the  apparent  difference  of 
declination  of  the  stars  is 

8'  -  8  =  401".13. 


RING   MICROMETER  239 

From  (342)  and  (343)  we  find 

A  =18°!'  34",  B  =  3°  55'  37"; 

and  then,  from  (344) 

r  =  211"  A. 

The   mean   value   of    r  from    nine    complete    sets   of 

transits  is 

r  =  210".7  ±  0".18. 

159.  To  determine  the  difference  of  the  right  ascensions 
and  declinations  of  two  stars.  Observe  the  transits  over 
the  edge  of  the  ring,  as  in  §  158.  Using  the  notation  of 
§  158,  the  difference  of  the  right  ascensions  is  given  by 

a'-a  =  i(V+-'s')-K'i+<8)-  (345) 

Letting  OM=  d,  and  OM'  =  df  we  can  write 
siny  =  3t2»«fc  _  ,,),      si,,  y  = 


c?  =  r  cos  y,  df  =r  cos  y'.  (347) 

The  difference  of  the  declinations  is  given  by 

B'-B  =  d'  ±d.  (348) 

The  lower  sign  is  used  if  the  stars  are  observed  on  the 
same  side  of  the  center  of  the  ring.  Equations  (347)  do 
not  determine  the  signs  of  d  and  d',  but  there  will  never 
be  any  ambiguity  -if  the  observer  notes  the  positions  of  the 
two  stars  with  reference  to  the  center  of  the  ring. 

Differences  obtained  in  this  way  are  slightly  in  error  on 
account  of  refraction  ;  on  account  of  the  fact  that  the  paths 
of  the  stars  are  arcs  and  not  chords  of  the  circle  (except 
for  equatorial  stars)  ;  and  on  account  of  the  proper  motion 
of  one  of  the  bodies  (if  it  have  a  proper  motion).  But  as 
stated  above,  the  ring  micrometer  should  not  be  used  on 
an  equatorial  telescope  when  exact  measurements  are 
required;  so  that  the  corrections  for  these  errors  will 
seldom  be  justified,  and  we  shall  not  consider  them  here. 


240  PRACTICAL  ASTRONOMY 

Example.  Saturday,  1888  Sept.  8.  The  position  of 
Comet  c  1888  was  compared  with  that  of  the  star  13\  1197 
in  Weisse's  BesseVs  Catalogue,  by  observing  the  transits  of 
the  star  and  comet  over  the  inner  edge  of  the  ring  microm- 
eter of  the  12J-inch  equatorial  of  the  Detroit  Observatory. 
The  times  of  transit  were 

Star  Comet 

*!  =  19*  18™  34M,'  </  =  19*  19™  12-.6, 

*2  =  19   18    50.0,  *a'  =  19   19    31.1. 

The  image  of  the  star  was  north  of  the  center  and  that  of 
the  comet  was  south.     The  radius  of  the  ring  is  171".6. 
The  apparent  place  of  the  star  was 

a  =  13*  56™  3-.06,  8  =  +  31°  13'  49".6. 

The  declination  of  the  comet  was  approximately  4-  31°  18'. 
Substituting  in  (345)  we  find 

a'  -  a  =  +  0™  398.80. 

The  solution  of  (346)  and  (347)  is  here  given. 

logJ£  0.87506  log-V-  0.87506 

cos  8  9.93201  cos  8'  9.93169 

a.c.  log r  7.76548  a.c.  log r  7.76548 

log  ('2  ~  'i)    1-20140              log  (t2f  -  V)  1.26717 

siny   9.77395  siny'  9.83940 

cosy  9.90542  cosy'  9.85912 

logr  2.23452  logr  2.23452 

d     138".0  d'  124".l 

8' -8  =  </'  +  </  =  +  262".l  =  +  4'  22".l 

These  approximate  values  of  the  apparent  differences 
a'  —  a  and  &'  —  B  correspond  to  the  Ann  Arbor  sidereal 
time  1888  Sept.  8, 19*  19™  22s. 


APPENDICES 


A.    HINTS  ON  COMPUTING 

The  numerical  calculations  required  in  the  problems  of 
practical  astronomy  are  generally  a  source  of  discour- 
agement to  the  beginner,  even  though  he  is  a  skilful 
mathematician.  Practice  in  making  extensive  series  of 
computations,  however,  very  soon  suggests  to  him  various 
devices  for  avoiding  much  of  the  labor.  Every  computer 
acquires  methods  peculiarly  his  own;  yet  the  following 
hints  will  possibly  be  useful  to  many. 

Only  those  logarithmic  tables  should  be  employed  which 
contain  the  auxiliary  tables  of  proportional  parts  on  the 
margins  of  the  pages,  excepting  possibly  three-place  and 
four-place  tables.  They  enable  the  computer  to  make 
nearly  all  the  interpolations  mentally ;  and  the  use  of  any 
other  tables,  for  any  purpose  whatever,  cannot  be  recom- 
mended. 

The  following  are  recommended : 

Bruhns's  or  Vega's  seven-place  tables. 

Bremiker's  six-place  tables. 

Hussey's,  NewcomVs  or  Becker's  five-place  tables. 

Zech's  addition  and  subtraction  logarithmic  tables.* 

Barlow's  tables  of  squares,  square-roots,  etc. 

Crelle's  multiplication  and  division  tables. 

If  extensive  computations  are  made  with  seven-place 
tables  and  the  interpolations  carried  to  hundredths  of  sec- 

*  Bremiker's  tables  contain  Gauss's  addition  and  subtraction  loga- 
rithmic tables  to  six  places.    Hussey's  and  Becker's  contain  Zech's  to 
five  places,  and  Newcomb's  contain  Gauss's  to  five  places. 
R  241 


242  PRACTICAL   ASTRONOMY 

onds,  the  results  are  usually  accurate  within  a  tenth  of  a 
second.  If  six-place  tables  are  used  and  the  interpolations 
carried  to  tenths  of  seconds,  the  results  are  usually  accu- 
rate within  a  second.  If  five-place  tables  are  employed 
and  the  interpolations  carried  to  seconds  or  to  hundredths 
of  minutes,  the  results  are  usually  accurate  within  five 
seconds. 

First  of  all)  an  outline  of  the  whole  solution  should  be 
prepared  by  writing  in  a  vertical  column  the  symbols  of 
all  the  functions  that  will  be  used.  Those  should  be 
placed  adjacent  to  each  other  which  are  to  be  combined, 
as  shown  by  the  formulae.  If  a  number  of  similar  solu- 
tions for  different  values  of  the  variable  or  variables  are  to 
be  made,  a  vertical  column  should  be  arranged  for  each  on 
the  right  of  the  column  of  functions,  which  thus  serves  for 
all,  and  the  computations  in  the  several  columns  should  be 
carried  on  simultaneously.  If  the  solutions  are  made  for 
equidistant  values  of  the  variable  or  variables,  this  method 
affords  a  valuable  check  on  the  accuracy  of  the  results, 
since  all  the  quantities  which  are  in  the  same  horizontal 
line  should  differ  systematically  from  each  other  as  we  go 
from  the  first  column  to  the  last.  By  subtracting  the 
result  in  each  column  from  the  corresponding  result  in 
the  next  column  to  the  right,  any  error  will  be  detected 
very  quickly  by  the  fact  that  the  differences  will  not  vary 
properly.  This  method  is  called  the  method  of  differences. 
It  will  not  detect  systematic  errors :  that  is,  errors  affect- 
ing all  the  columns  alike. 

If  the  sine,  cosine,  tangent,  etc.,  of  the  same  angle  are 
required,  they  should  all  be  taken  from  the  tables  at  one 
opening.  Avoid  turning  twice  to  the  same  angle  in  the  same 
solution.  The  interpolations  can  be  checked  by  subtract- 
ing mentally  the  last  two  figures  of  the  cosine  from  those 
of  the  sine  and  comparing  the  result  with  the  last  two 
places  of  the  tangent;  and  similarly  in  other  cases. 

The  tangent  of  an  angle  always  varies  more  rapidly 


APPENDICES  243 

than  its  sine  or  cosine,  and  for  this  reason  the  value  of  an 
angle  should  be  taken  from  the  tables  by  means  of  its  tan- 
gent if  great  accuracy  is  required. 

Many  of  the  operations  can  be  performed  mentally, 
thereby  saving  much  time.  Thus,  two  numbers  can  be 
added  or  subtracted  mentally  from  left  to  right^  or  a  num- 
ber multiplied  or  divided  by  two  from  left  to  right,  and 
the  result  held  in  mind  while  we  turn  to  the  tables  and 
take  out  the  proper  angle  or  function.  This  has  been 
done  very  largely  in  the  solutions  of  the  examples  in  this 
book.  Just  how  far  the  student  should  carry  the  method 
depends  upon  the  individual.  The  beginner  will  find  it 
perplexing  arid  a  fruitful  source  of  error,  but  after  some 
practice  he  can  perform  the  operations  quickly  and  accu- 
rately. It  should  be  said  that  many  experienced  com- 
puters prefer  to  set  down  the  results  in  the  usual  way. 

If  there  are  several  factors  in  the  numerator  or  denomi- 
nator of  an  expression  to  be  evaluated,  do  not  add  the 
logarithms  in  the  numerator  together,  those  of  the  de- 
nominator together,  and  take  the  difference ;  but  form  the 
arithmetical  complement  of  each  logarithm  in  the  denomi- 
nator mentally  by  subtracting  it  from  10,  from  left  to  right, 
and  set  down  the  result  in  the  proper  place.  All  the 
factors  can  then  be  combined  by  one  addition. 

When  a  constant  quantity  is  to  be  used  several  times,  it 
should  be  written  on  the  margin  of  a  slip  of  paper  and  held 
over  the  quantities  with  which  it  is  to  be  combined. 

If  two  quantities  are  to  be  combined  which  are  separated 
by  one  or  more  lines,  hold  a  pencil  or  slip  of  paper  over  the 
intervening  quantities  and  the  two  can  then  be  combined 
as  conveniently  as  if  they  were  adjacent. 

If  two  quantities  are  given  by  their  logarithms,  and  the 
logarithm  of  their  sum  or  difference  is  required,  it  should 
be  found  by  means  of  addition  and  subtraction  logarithmic 
tables.  The  result  will  be  obtained  more  quickly  and 
accurately  than  by  means  of  the  ordinary  tables. 


244  PRACTICAL   ASTKONOMY 

Whenever  the  formulae  furnish  checks  on  the  accuracy 
of  the  solution,  they  should  generally  be  applied.  The  ex- 
perienced computer  usually  detects  an  error  very  quickly. 

If  the  trigonometrical  function  or  other  logarithmic 
function  is  negative,  write  the  subscript  n  after  it. 

Do  not  use  negative  characteristics.  Increase  them  by  10 
if  they  are  naturally  negative. 

The  example  in  §  4  will  illustrate  many  of  these  methods. 
First  write  down  the  two  columns  of  functions,  as  the 
outline  of  the  solution  of  (12),  (13),  (14)  and  the  check 
equation  (9).  The  values  of  </>,  z  and  A  are  inserted. 
From  the  tables  tan  z  and  sin  z  are  found  and  written  oppo- 
site their  symbols ;  and  likewise  cos  J.,  tan  A  and  sin  A. 
The  sum  of  tan  z  and  cos  A  is  tan  M.  Add  them  mentally, 
enter  the  tables  and  take  out  M.  Take  out  sin  M  at 
once.  Subtract  Mirom  </>,  mentally,  find  sec  ((/>  —  M)  and 
tan  (cf>  —  M).  The  value  of  sec  (<£  —  M)  is  10  — cos  ($  —  M). 
Add  the  three  logarithms  to  find  tan  t.  Determine  t  from 
the  tables  and  take  cost  and  cosec£  out.  The  sum  of 
tan  (<£  —  M)  and  cos  t  is  tan  8.  Add  them  mentally  and 
take  8  and  sec  8  from  the  tables.  The  sum  of  the  last  four 
logarithms  is  log  1.  It  should  not  differ  more  than  one  or 
two  units  of  the  last  place  from  zero  or  ten. 

The  printed  solution  contains  everjr  figure  that  need  be 
written  down.  But  possibly  the  beginner  should  write 
down  tan  M,  <f>  —  M  and  tan  8. 

B.    INTERPOLATION  FORMULA 

The  Ephemeris  tabulates  the  values  of  any  required 
function  corresponding  to  equidistant  values  of  the  time. 
If  the  value  of  the  function  for  any  intermediate  date  is 
desired,  it  is  sometimes  determined  most  conveniently  by 
means  of  a  general  interpolation  formula. 

Let  T  be  the  date  in  the  Ephemeris  nearest  the  instant 
for  which  the  value  of  the  function  is  required,  and  let 
&)  be  the  tabular  interval  of  time.  Then  the  adjacent  dates 


APPENDICES 


245 


in  the  Ephemeris  may  be  represented  as  in  the  first  column 
of  the  following  table,  and  the  corresponding  values  of  the 
function  as  in  the  second  column.  Subtracting  each  value 


Argument 

Function 

1st  Diff. 

2dDiff, 

3d  Diff, 

4th  Diff, 

5th  Diff, 

6th  Diff, 

«•-•• 

/df-»*) 

ain 

T-2o> 

/(  T  —  2  CD) 

bn 

a,, 

ctf 

T7-     o> 

/(r-   <«>) 

b, 

rf, 

a, 

c, 

*/ 

T 

/(T7) 

M 

b 

w 

d 

M 

/ 

a' 

c' 

e' 

T  +     w 

/(r+   ») 

b' 

d' 

a" 

c" 

71  +  2  <o 

f(T+  2o>) 

b" 

a"1 

T+  3w 

/(r+3o>) 

of  the  function  from  the  next  following  value,  we  obtain 
the  "1st  differences"  in  column  three.  Subtracting  each 
1st  difference  from  the  next  following,  we  obtain  the  2d 
differences  in  column  four;  and  so  on,  for  the  differences 
of  higher  orders.  Lastly,  the  quantities  [#]  =  ^  (a,,  +  a'), 
[c]  =  J  (c,  +  c')  and  [e~]  =  ±(e,  +  e')  are  inserted  in  the 
same  horizontal  line  as  T.'  Let  the  instant  for  which  the 
value  of  the  function  is  wanted  be  represented  by  T  + 1. 
Let  n  be  the  ratio  of  t  and  the  tabular  interval  o>;  i.e., 
n  =  t/(o,  or  t  =  no).  The  value  of  the  function  required  is 


2-3 


Example.  Determine  from  the  American  Ephemeris, 
page  219,  the  apparent  declination  of  Mercury  at  Green- 
wich mean  time  1899  April  2d  16h  Om. 


246 


PRACTICAL   ASTRONOMY 


The  epoch  T  is  April  3rf.O,  the  tabular  interval  co  is  24\ 
and  t  is  —  Sh.  Therefore,  n  =  —  J.  The  functions  and 
differences  are  as  below: 


Argument 

Function 

a 

b 

c 

d 

e 

March  3HO 
April     1  .0 
2  .0 

+  13°  16  '34  "A 
13  22  29  .6 
13  24  29  .6 

+  5'55".2 
+  2     0  .0 
-1  54  .4 

-3'55".2 
-3  54  .4 

+0".8 
+  3  .2 

+2".4 

+  0".3 

3.0 

13  22  35  .2 

[-3  50  .0] 

-3  51  .2 

[+4  .6] 

+  2  .7 

[  +  0  .1] 

4.0 
5.0 
6.0 

13  16  49  .6 
13     7  18  .7 
+  12  54  11  .0 

-5  45  .6 
-9  30  .9 

-13     7  .7 

-3  45  .3 
-3  36  .8 

+5  .9 
+  8  .5 

-1-2  .6 

-0  .1 

The  quantities  to  be  taken  from  this  table  are  all  in- 
cluded, with  April  3d.O,  between  the  two  horizontal  lines. 
Substituting  these  values  and  n  =  —  J  in  the  above  equa- 
tion we  obtain  the  following  values  of  the  individual 
terms,  and  their  sum,  respectively : 

+  13°  22'  35".2 

+          1  16  .7 

12  .8 

-f  0  .2 

0  .0 

0  .0 


+  13°  23'  39".3 


It  is  often  required  to  determine  by  interpolation  the 
value  of  a  function  for  a  date  midway  between  two  tabu- 
lar dates.  The  required  value  is  determined  as  follows  : 
From  the  arithmetical  mean  of  the  two  values  of  the 
function  corresponding  to  the  two  tabular  dates,  subtract 
one-eighth  of  the  arithmetical  mean  of  the  second  differ- 


APPENDICES  247 

ences  found  on  the  same  horizontal  lines  as  the  two  dates, 
and  add  three  one  hundred  and  twenty-eighths  of  the 
arithmetical  mean  of  the  fourth  differences  found  on  the 
same  horizontal  lines. 

Example.     Determine  the  apparent  decimation  of  Mer- 
cury at  Greenwich  mean  time  1899  April  3d.5. 
From  the  above  data 

i  (13°  22'  35".2  +  13°  16  49".6)  =  +  13°  19'  42".4 
_   |   .£(_      351.2-          345.3)  =  +  28.5 

+  T|¥.^(+  2.7+  2.6)= 0  .0 

App.  S,  1899  April  3<*.5  =  +  13°  20'  10".9 

C.  COMBINATION  AND  COMPARISON  OF  OBSERVATIONS 

Formulae  resulting  from  the  Method  of  Least  Squares 

1.   Direct  observations  of  a  quantity :  n  separate  results, 
mv  m^,  •••  mn  of  equal  weight. 

r    T  & 
Most  probable  value  of  quantity,  z  =  ^— =*• 

Residuals,  z  —  ml  =  vv  z  —  m2  =  r2,  •••  z  —  mn  =  vw 
Probable  error  of  z,  r0  =  ±  0.6745 


Probable  error  of  a  single  observation,    r  =  ±  0.6745 

\n  -  1 

2.   Direct  observations  of  a  quantity :  n  separate  results, 
mv  mv  •••  mn  of  unequal  weights,  pv  pv  •••  pn. 

Most  probable  value  of  quantity,  z  =  y      • 


Probable  error  of  z,  .  r0  =  ±  0.6745 

Probable  error  of  an  obs'n  of  weight  unity,  r  =±  0.6745  .. 

*  The  symbols  [  ]  signify  the  sum  of  all  similar  quantities.    Thus 

LWIJ  EE  771 1  -f-  7712  +  "•  +  fftn. 


248  PRACTICAL  ASTRONOMY 

Weight  of  z,  P  =  \_p]. 

Relation  of  weights  to  probable  errors,       ^j  : p2  :..•::  — 


3.  -  If  Z  =  azl  ±  bz2  ±  •••  kzn,  and  the  probable  errors  and 
weights  of  zv  z2,  •••  zn  are  rv  rv  •••  rn  and  pv  p2,  —  pn,  then 
the  probable  error  and  weight  of  Z  are  given  by 


P       Pi       P2  P 


4.   In  general,  if  Z=f(z^  z2,  •••  2n),  the  probable  error 
of  Zis 


5.  Direct  observations  of  a  function  of  a  quantity  z: 
the  separate  results,  mv  m2,  •••  mn  of  equal  weight,  and 
the  form  of  the  function,  az.  The  observation  equations 
are 

cijZ  +  ml  =  0, 
agZ  +  m2  =  0, 


anz  +  mn  =  0. 
The  most  probable  value  of  z  and  its  probable  error  are 

—  [=?•  r  =  ±0-6745V[aa][(I]-l)' 

If  the  observations  are  of  unequal  weights,  multiply  the 
observation  equations  through  by  the  square  roots  of  their 
respective  weights,  and  proceed  as  before. 

6.  Direct  observations  of  a  function  of  two  quantities, 
w  and  z :  the  separate  results,  mv  mv  .  .  .  mn  of  equal 
weights,  and  the  form  of  the  function,  aw  +  bz.  The  ob- 
servation equations  are 

a^w  +  b^z  +  ml  =  0, 
a2w  +  £>2z  +  7«2  =  0, 

OnW  +  bn£  +  mn  =  0. 


APPENDICES  249 

The  normal  equations  are 

[aa]  w  +  [a6]  z  +  [am]  =  0, 
[a6]  w  +  [66]  z  +  [6m]  =  0. 
Let 

[66]  -  g]  [aft]  =,  [66.1],         [6m]  -  jgj  [am]  =  [6m.l]. 
Then  the  most  probable  values  of  w  and  z  are  given  by 


[66.1]' 


[aa]         [aa] 
The  weights  of  w  and  2  are 


The  probable  error  of  a  single   observation  (of  weight 
unity)  is 


and  the  probable  errors  of  w  arid  z  are 


If  the  observations  are  of  unequal  weights,  multiply  the 
observation  equations  through  by  the  square  roots  of  their 
respective  weights,  and  proceed  as  before. 

7.  Direct  observations  of  a  function  of  three  quantities, 
x,  y  and  z  :  the  separate  results,  mv  m^  .  .  .  mn  of  equal 
weight,  and  the  form  of  the  function,  ax  -f  by  +  cz.  The 
observation  equations  are 

OjX  +  bjy  +  Cj3  +  r^  =  0, 
a^  -f  6^  +  c2z  +  m2  =  0, 


mn  =  0. 
The  normal  equations  are 

[aa]  x  +  [a6]  y  +  [ac]  z  +  [am]  =-,  0, 
[a&]  a:  -f  [66]  y  +  [6c]  z  +  [6m]  =  0, 
[ac]  x  +  [6c]  y  +  [afj  s  +  [cm]  =  0. 


250  PRACTICAL  ASTRONOMY 

Let 


M[a"3  =c*m'1]' 

-  £]  M       =  [<*•!],        [«.]     -  gj  [am]      =  [em.1], 

[^  -  prj  [^  =  E^].     [-•!]  -  g^  P»J]  =  [--2]- 

Then  the  most  probable  values  of  a?,  y  and  z  are  given  by 


- 


[66.1]          [66.1]' 

_[o6]        __[«£]'  [am] 
[aa]    ^       [aa]Z  * 

The  weights  of  #,  «/  and  2  are  given  by 
P.  =  [<*.2], 


.  hM 

in  which 


The  probable  error  of  a  single   observation  (of  weight 
unity)  is 


r  =  ±  0.6745 


and  the  probable  errors  of  #,  y  and  2  are 


„,,—£:,       r,=    r          P.=_!L. 


If  the  observations  are  of  unequal  weights  multiply  the 
observation  equations  through  by  the  square  roots  of  their 
respective  weights,  and  proceed  as  before. 


APPENDICES 


251 


D.   OBJECTS  FOR  THE  TELESCOPE  • 

Besides  the  moon,  the  planets  and  the  Milky  Way,  the 
objects  in  the  following  list  will  be  of  interest  to  the  stu- 
dent. Fuller  descriptions  of  them,  with  many  valuable 
hints  on  the  use  of  the  telescope,  can  be  found  in  Webb's 
Celestial  Objects  for  Common  Telescopes,  which  is  an  excel- 
lent guide  for  the  observer.  Every  student  should  provide 
himself  with  a  good  star  atlas.  Klein  s  Star  Atlas,  or  Heis's 
Atlas  Coelestis  is  recommended. 


a,  1900.0 

5,  1900.0 

Object:  description:  remarks 

0*  37*.  3 

+  40°  43' 

The  Great  Nebula  in  Andromeda.     One  of 

the  most  interesting  in  the  sky,  large, 

2£°  by  4°,  easily  visible  to  the  naked  eye. 

A  small  companion  nebula  lies  22'  south. 

0    53  .4 

+  81     20 

U  Cephei,  variable,  lm.l  to  9W.2,  period  2<*.5. 

1     18  .9 

+  67     36 

\l/  Cassiopeia,  triple,  ^4TO.5,  B9m,  O10m. 

AB  =  30",  BC  =  3". 

1    22   .6 

+  88    46 

a  Ursce  Minoris  or  Polaris,  the  standard  2™ 

star  ;   a  9W  companion  at  s  =  18".  5. 

1     48   .0 

+  18    48 

7  Arietis,  double,  4W.5  and  5m,  p  =  179°, 

s  =  8". 

1     57   .7 

+  41    51 

7  Andromeda:,  double,  3m.5  and  5m.5,  p  = 

63°,  s  =  10".     The  5™.  5  is  also  double, 

but  close  and  difficult  even  for  the  largest 

telescopes. 

2     12   .0 

+  56    41 

Cluster  in  Perseus.     A  magnificent  object 

with  a  low  power.     Another  fine  cluster 

3  minutes  east. 

2     14   .3 

-    3    26 

o  Cetii  interesting  variable,  irregular,  1«*.7 

to  9m.5,  period  about  33K 

3      1    .7 

+  40    34 

/3  Persei  (Algol),  interesting  variable,  2W.3 

to  3m.5,  period  2<*  20*  48"  55V 

3    40   .2 

+  23    27 

Nebula  in  the  Pleiades,  very  faint  and  diffi- 

cult, Merope  in  its  north  extremity. 

4      7   .6 

+  50    59 

Cluster  in  Perseus,  good  with  low  power. 

4      9  .6 

-  13      0 

Planetary  nebula  in  Eridanus,  circular,  12"* 

star  in  cejiter. 

4    30   .2 

+  16     19 

a  Tauri  (Aldebaran),  lm  star,  red. 

5      9  .3 

+  4.5     54 

a  Aurigce  (Capella),  lm  star. 

5      9  .7 

-    8     19 

/3  Orionis  (Rigel),  double,  lm  and  9m,  s  = 

9".  5.     The  9m  is  a  close  double,  very  dif- 

ficult even  with  the  largest  instruments. 

5    28   .5 

+  21     57    j 

Nebula  in  Taurus  large,  faint,  oblong. 

252 


PRACTICAL   ASTRONOMY 


o,  1900.0 

5,  1900.0 

Object:  description:  remarks 

5*  30"*.  4 

-    5°  27' 

The  Great  Nebula  in  Orion,  one  of  the 

most    interesting    nebulae    in   the    sky. 

about  3°  by  5°  in  size.     Near  its  densest, 

part  is  the  multiple  star  6  Orionis,  called 

the  Trapezium.     The  spectrum  of  the 

nebula  indicates  a  gaseous  composition. 

5    35   .7 

-    2      0 

f  Orionis,  triple,  A  3™,  B6m.5,  C  10™,  AB  = 

2".5,  ^1(7  =  57". 

6      2   .7 

+  24    21 

Cluster  in  Gemini,  fine  field  with  low  power. 

6    37   .4 

+  59    33 

12  Lyncis,  triple,  A  6m,  B6m.5,  C7m.5,  AB  = 

6    40   .7 

-  16    34 

a  Canis  Majoris  (Sirius},  the  brightest  star 

in  the  sky.    A  close  10  mag.  companion  is 

now  (1899)  difficult  in  powerful  tele- 

scopes. 

7     14  .1 

+  22     10 

d  Geminorum,  double,  one  yellow  3ra.5,  the 

other  red  Sm,  p  =  205°,  s  =  7". 

7     28   .2 

+  32      6 

a  /Geminorum  (Castor},  fine  double,  3™  and 

/  3ro.5,  p  -  226°,   s  =  5".  7. 

7     34   .1 

+    5    29 

a  Canis  Minoris  (Procyon}  lm,  with  13"» 

companion  discovered  in  1896,  difficult 

with  large  instruments.     At  discovery, 

p  =  320°,  s  =  4".  7. 

8    34   .5 

+  20     17 

Cluster  in  Cancer  (Prcesepe},  fine  field  with 

-low  power. 

8    45   .7 

+  12     10 

Cluster  in  Cancer,  about  200  stars,  9M  to  15m. 

9    47    .2 

+  69    36 

Nebulae,  in   Ursa  Major,  two  nebulae,  30' 

apart,  preceding  one  brighter  with  bright 

nucleus. 

10    14   .4 

+  20    21 

7  Leonis,  fine  double,  2W  and  3m.  5.    In  1897, 

p  =  115°,  s  =  3".8. 

10    19   .9 

-17     39 

Planetary  Nebula  in  Hydra,  fairly  bright. 

11     12   .5 

+  59    19 

Nebula  in  Ursa  Major,  small,  bright,  with 

nucleus. 

11     47   .7 

+  37    33 

Nebula  in  Ursa  Major,  bright,  3'  to  4'  in 

• 

diameter. 

12       5   .0 

+  19      6 

Cluster  in  Coma  Berenices,  globular,  bright, 

well  resolved  in  large  telescopes. 

12     34   .8 

-  11      4 

Nebula  in  Virgo,  elliptical,  30"  by  5',  fine 

field  with  low  power. 

12     36   .6 

-    0    54 

7   Virginis,  double,  4TO  and  4W.     In  1898, 

p  =  330°,  s  =  6". 

13     19    .9 

+  55    27 

f  Ursce  Majoris,  fine  double,  3™  and  5TO, 

.s  =  14". 

13    37   .5 

+  28     52 

Cluster  in  Canes  Venatici,  bright,  globular, 

probably  more  than  1,000  stars. 

14     11    .1 

+  19    43 

a  Bootis  (Arcturus),  1m  star,  yellow. 

APPENDICES 


253 


a,   1900.0 

5,  1900.0 

Object:  description:  remarks 

S 

14*  40M.6 

+  27°  30' 

e  .Book's,  beautiful  double,  3m  yellow  and  lm 

blue,  s  =  3",  p  =  328°. 

15     14   .1 

+  32       1 

U  Corona',  variable,  7m.5  to  8m.9,  period  3d 

10*  51"'. 

16     23   .3 

-  26     13 

a  Scorpii,  double,  lm  and  7m,  s  =  3". 

16     37   .5 

+  31     47 

f  Herculis,  double,  3™  and  6W,  s  -  0".6  in 

1899,  period  about  35  years,  now  (1899) 

very  difficult  in  large  instruments. 

^ 

16     38   .1 

+  36     39 

The  Cluster  in  Hercules,  globular,  one  of 
the  finest  of  its  kind.        .                  [eter. 

16    40   .3 

+  23     59 

Nebula  in  Hercules,  planetary,  8"  in  diam- 

17     10   .1 

+  14     30 

a  Herculis,  variable,  3"».l  to  3W.  9,  irregular 

V- 

period  ;    companion    5m.5   at   p  =  116°, 
*  =  4".7. 

17     11    .5 

+    1     19 

U  Ophiuchi,  6m.O  to  6«7,  period  20*  8m. 

17     51    .1 

-18     59 

Cluster  in  Ophiuchus,  good  field  with  low 

power. 

>y 

17     58   .6 

+  66     38 

Nebula  in  Draco,  planetary,  bright,  diam- 

eter 35",  very  near  pole  of  ecliptic,  very 

interesting. 

18      7   .3 

+    6     50 

Nebula    in    Ophiuchus,   planetary,   bright, 

diameter  5". 

18    33   .6 

+  38    41 

a  Lyrce  (Vega),  brightest  star  in  northern 

hemisphere. 

'!\A 

18    41   .0 

+  39    34 

e  Lyrce,  a  multiple  star,  A  5™,  B  6W.5,  C  5™, 

r^ 

D  5™.5,  AB  3",  CD  2".3,  AC  207".    Nu- 

merous small  stars  between  AB  and  CD. 

S*~ 

18    46   .4 

+.33     15 

ft  Lyrce,  variable,  3m.4  to  4"*.5,  period  12d 

21*  47m 

^k. 

18    49   .8 

+  32     54 

Ring  Nebula  in  Lyra,  annular,  gaseous, 

r*1 

•^ 

most  interesting  of  its  kind. 

~Q 

19    26   .7 

+  27    45 

/3  Cygni,  fine  double,  3m  yellow  and  7m  blue, 

fs 

, 

p  =  56°,  s  =  35"'. 

19    48   .5 

+  70      1 

6  Draconis,  double,  5"*.  5  and  7m.5,  s  =  3". 

19    .55   .2 

+  22     27 

Nebula  in  Vtilpecula,  the  "  Dumb  Bell  Neb- 

ula," double,  large. 

20    42   .0 

+  15     46 

7  Delphini,  double,  4"1  and  6m,  s  =  11". 

20     58   .7 

-  11     45 

Nebula  in  Aquarius,  planetary,  bright,  very 

^L 

-21  '    2^4 

+  38     15 

interesting  in  a  large  telescope. 
61  Cygni,  double,  5».5  and  6M,  s  =  21",  one 

of  the  nearest  stars  to  us. 

21       8   .2 

+  68      5 

T  Cephei.  variable,  S^.O  to  9W.9,  period  383d. 

v 

21     28   .2 

-    1     16 

Cluster  in  Aquarius,  large,  globular.      |__^- 

22    23   .7 

-    0    32 

f  Aquarii,  double,  4m  and  4m.5,  s  =  3'  '.3  in 

1897. 

23     21    .1 

+  41     59 

Nebula   in   Andromeda,  planetary,  small, 

very  bright,  round. 

254 


PRACTICAL    ASTRONOMY 


TABLE  I.    PULKOWA  REFRACTION  TABLES 


App't 
z 

log/* 

A 

App't 
z 

logM 

A 

App't 

z 

log  /a. 

A 

A 

o  / 

0    I 

0    / 

0  0 

1.76032 

71  0 

1.75614 

1.0115 

77  0 

1.75131 

.0253 

1.0029 

5  0 

1.76032 

10 

1.75606 

1.0118 

10 

1.75107 

.0259 

1.0029 

10  0 

1.76030 

20 

1.75598 

1.0120 

20 

1.75083 

.0264 

1.0030 

15  0 

1.76028 

30 

1.75590 

1.0123 

30 

1.75058 

.0271 

1.0030 

20  0 

1.76025 

40 

1.75582 

1.0125 

40 

1.75032 

.0278 

1.0031 

25  0 

1.76021 

50 

1.75573 

1.0128 

50 

1.75005 

.0285 

1.0032 

30  0 

1.76015 

72  0 

1.75564 

1.0130 

78  0 

1.74976 

1.0293 

1.0033 

35  0 

1.76006 

10 

1.75555 

1.0133 

10 

1.74947 

1.0300 

1.0033 

40  0 

1.75995 

20 

1.75546 

1.0136 

20 

1.74917 

1.0309 

1.0034 

45  0 

1.75980 

1.0018 

30 

1.75536 

1.0138 

30 

1.74886 

1.0318 

1.0035 

50  0 

1.75960 

1.0022 

40 

1.75526 

1.0141 

40 

1.74853 

1.0327 

1.0036 

51  0 

1.75955 

1.0024 

50 

1.75516 

1.0144 

50 

1.74819 

1.0335 

1.0037 

52  0 

1.75949 

1.0025 

73  0 

1.75506 

1.0147 

79  0 

1.74783 

1  .0344 

1.0038 

53  0 

1.75943 

1.0026 

10 

1.75496 

1.0150 

10 

1.74746 

1.0354 

1.0039 

54  0 

1.7,5936 

1.0027 

20 

1.75485 

1.0153 

20 

1.74707 

1.0364 

1.0040 

55  0 

1.75928 

1.0029 

30 

1.75474 

'  .0157 

30 

1.74665 

1.0374 

1.0041 

56  0 

1.75920 

1.0032 

40 

1.75462 

.0160 

40 

1.74623 

1.0385 

1.0042 

57  0 

1.75912 

1.0035 

50 

1.75450 

.0163 

50 

1.74579 

1.0397 

1.0043 

58  0 

1.75902 

1.0038 

74  0 

1.75438 

.0166 

80  0 

1.74533 

1.0409 

1.0044 

59  0 

'1.75892 

1.0041 

10 

1.75425 

1.0170 

10 

1.74484 

1.0421 

1.0045 

60  0 

1.75881 

1.0044 

20 

1.75412 

1.0173 

20 

1.744*33 

1.0433 

1.0046 

61  0 

1.75868 

1.0047 

30 

1.75398 

1.0177 

30 

1.74380 

1.0447 

1.0048 

62  0 

1.75853 

1.0051 

40 

1.75384 

.0181 

40 

1.74325 

1.0461 

1.0049 

63  0 

1.75837 

1.0055 

50 

1.75369 

1.0185 

50 

1.74266 

1.0475 

1.0050 

64  0 

1.75820 

1.0059 

75  0 

1.75354 

1.0188 

81  0 

1.74204 

1.0491 

1.0052 

65  0 

1.75801 

1.0064 

10 

1.75338 

1.0191 

10 

1.74139 

1.0508 

1.0053 

66  0 

1.75780 

1.0070 

20 

1.75322 

1.0195 

20 

1.74071 

1.0525 

1.0055 

67  0 

1.75755 

1.0077 

30 

1.75306 

1.0200 

30 

1.73999 

1.0542 

1.0057 

68  0 

1.75727 

1.0085 

40 

1.75289 

.0205 

40 

1.73924 

1.0561 

1.0059 

69  0 

1.75694 

1.0093 

50 

1.75271 

.0211 

50 

1.73844 

1.0580 

1.0061 

70  0 

1.75657 

1.0103 

76  0 

1.75253 

.0216 

82  0 

1.73760 

1.0600 

1.0063 

10 

1.75650 

1.0105 

10 

1.75235 

.0223 

10 

1.73671 

1.0622 

1.0065 

20 

1.75643 

1.0107 

20 

1.75216 

.0229 

20 

1.73577 

1.0645 

1.0008 

30 

1.75636 

1.0109 

30 

1.75196 

.0235 

30 

1.73478 

1.0669 

1.0070 

40 

1.75629 

1.0111 

40 

1.75175 

.0241 

40 

1.73373 

1.0694 

1.0073 

50 

1.75622 

1.0113 

50 

1.75153 

.0246 

50 

1.73260 

1.0720 

1.0076 

71  0 

1.75614 

1.0115 

77  0 

1.75131 

1.0253 

83  0 

1.73143 

1.0747 

1.0078 

SUPPLEMENT 


App't 
z 

log 
Mtanz 

X 

A 

App't 
z 

log 
Mtan  z 

X 

A 

0    / 

0   I 

8230 

2.61534 

1.0669 

1.0070 

8630 

2.88535 

1.1934 

1.0203 

83  0 

2.64226 

1.0747 

1.0078 

87  0 

2.93113 

1.2277 

1.0241 

8330 

2.67076 

1.0839 

1.0087 

8730 

2.98087 

1.2708 

1.0294 

84  0 

2.70088 

1.0949 

1.0098 

88  0 

3.03519 

1.3241 

1.0357 

8430 

2.73294 

1.1080 

1.0112 

8830 

3.09458 

1.3902 

1.0437 

85  0 

2.76717 

1.1235 

1.0127 

89  0 

3.15994 

1.4729 

1.0541 

8530 

2.80376 

1.1424 

1.0148 

8930 

3.23206 

1.5762 

1.0680 

86  0 

2.84304 

1.1652 

1.0172 

90  0 

3.31186 

1.7046 

1.0859 

APPENDICES 


255 


TABLE  I.    PULKOWA  REFRACTION  TABLES 
B.     Factor  depending  on  the  Barometer 


English 
inches 

log£ 

English 
inches 

log  B 

French 
metres 

log  B 

25.0 

-  0.07330 

28.0 

-  0.02409 

0.724 

-  0.01621 

25.1 

-0.07157 

28.1 

-  0.02254 

0.726 

-  0.01500 

25.2 

-  0.06984 

28.2 

-  0.02099 

0.728 

-0.01380 

25.3 

-  0.06812 

28.3 

-  0.01946 

0.730 

-0.01261 

25.4 

-  0.06641 

28.4 

-  0.01793 

0.732 

-0.01142 

25.5 

-  0.06470 

28.5 

-  0.01640 

0.734 

-  0.01024 

25.6 

-  0.06300 

28.6 

-  0.01488 

0.736 

-  0.00906 

25.7 

-  0.06131 

28.7 

-  0.01336 

0.738 

-  0.00788 

25.8 

-  0.05962 

28.8 

-0.01185 

0.740 

-  0.00670 

25.9 

-  0.05794 

28.9 

-  0.01035 

0.742 

-  0.00553 

26.0 

-  0.05627 

29.0 

-  0.00885 

0.744 

-  0.00436 

26.1 

-  0.05640 

29.1 

-  0.00735 

0.746 

-  0.00319 

26.2 

-  0.05294 

29.2 

-  0.00586 

0.748 

-  0.00203 

26.3 

-  0.05129 

29.3 

—  0.00438 

0.750 

-  0.00087 

26.4 

-  0.04964 

-29.4 

-  0.00290 

0.752 

+  0.00028 

26.5 

-  0.04800 

29.5 

-  0.00142 

0.754 

+  0.00144 

26.6 

-  0.04636 

29.6 

+  0.00005 

0.756 

+  0.00259 

26.7 

-  0.04473 

29.7 

0.00151 

0.758 

+  0.00374 

26.8 

-  0.04311 

29.8 

0.00297 

0.760 

+  0.00488 

26.9 

-  0.04149 

29.9 

0.00443 

0.762 

+  0.00602 

27.0 

-  0.03988 

30.0 

0.00588 

0.764 

+  0.00716 

27.1 

-  0.03827 

30.1 

0.00732 

0.766 

+  0.00830 

27.2 

-  0.03667 

30.2 

0.00876 

0.768 

+  0.00943 

27.3 

-  0.03508 

30.3 

0.01020 

0.770 

+  0.01056 

27.4 

-  0.03349 

30.4 

0.01163 

0.772 

+  0.01168 

27.5 

-  0.03191 

30.5 

0.01306 

0.774 

+  0.01281 

27.6 

-  0.03033 

30.6 

0.01448 

0.776 

+  0.01393 

27.7 

-  0.02876 

30.7 

0.01589 

0.778 

+  0.01505 

27.8 

-  0.02720 

30.8 

0.01731 

0.780 

+  0.01616 

27.9 

-  0.02564 

30.9 

0.01871 

0.782 

+  0.01727 

28.0 

-  0.02409 

31.0 

+  0.02012 

0.784 

+  0.01837 

T.     Factor  depending  on  Attached  Thermometer 


Pahr, 

log  T 

Cent. 

log  T 

-20° 

+  0.00201 

-  30° 

+  0.00209 

-10 

0.00162 

-25 

0.00174 

0 

0.00123 

-20 

0.00139 

+  10 

0.00085 

-15 

0.00104 

20 

0.00047 

-10 

0.00069 

30 

+  0.00008 

-  5 

+  0.00035 

40 

-  0.00030 

0 

0.00000 

50 

-  0.00069 

+  6 

-  0.00035 

60 

-  0.00108 

10 

-  0.00069 

70 

-  0.00146 

15 

-  0.00104 

80 

-  0.00184 

20 

-  0.00138 

90 

-  0.00222 

25 

-0.00173 

+  100 

-  0.00262 

+  30 

-  0.00207 

256 


PRACTICAL   ASTRONOMY 


TABLE  I.     PULKOWA  REFRACTION  TABLES 
7.    Factor  depending  on  External  Thermometer 


Fahr. 

log  7 

Fahr, 

log  7 

Cent, 

log  7 

—  22° 

+  0.06560 

+  35° 

+  0.01200 

—  30° 

+  0.0(55(50 

-21 

0.06461 

36 

0.01112 

—  29 

0.0(5:581 

-20 

0.06361 

37 

0.01023 

—  28 

0.06202 

-19 

0.062(52 

38 

0.00935 

—  27 

0.0(>023 

-18 

0.06162 

39 

0.00848 

—  26 

0.0.r846 

-17 

0.06063 

40 

0.00760 

-  25 

0.05(5(59 

-10 

0.05964 

41 

0.00672 

-24 

0.05493 

—  15 

0.05866 

42 

0.00585 

-23 

0.05317 

—  14 

0.05767 

43 

0.0045)8 

-22 

0.05142 

-13 

0.05669 

44 

O.OQ411 

—  21 

6.049(58 

-12 

0.05571 

45 

0.00324 

-20 

0.04795 

—  11 

0.05473 

46 

0.00238 

-19 

0.04(522 

-10 

0.05376 

47 

O.OOlol 

—  18 

0.04451 

—  9 

0.05279 

48 

+  0.000*  14 

—  17 

0.04279 

—  8 

0.05182 

49 

—  0.00022 

—  16 

0.04108 

—  7 

0.05085 

50 

—  0.00107 

—  15 

0.03938 

—  6 

0.04988 
0.04891 

51 
52 

—  0.00193 
—  0.00279 

—  14 
—  13 

0.03769 
0.03(501 

^ 

0.04795 

53 

-  0.00364 

—  12 

0.03433- 

3 

0.04699 

54 

—  0.00449 

—  11 

0.03265 

—  2 

0.04(503 

55 

—  0.00535 

—  10 

0.030H9 

-  1 

0.04508 

56 

—  0.00(520 

—  9 

0.02933 

0 

0.04413 

57 

—  0.00704 

—  8 

0.027(57 

+  1 

0.04318 

58 

—  0.00789 

—  7  • 

0.02(502 

2 

0.04223 

59 

T-  0.00873 

—  6 

0.02438- 

3 

0.04128 

60 

—  0.00957 

—  5 

0.02274 

4 

0.04033 

61 

—  0.01041 

\ 

0.02112 

5 

0.03938 

62 

—  0.01125 

0 

0.019.r,0 

6 

0.03844 

63 

—  0.01209 

o 

0.01788 

7 

0.03750 

64 

—  0.01293 

-i 

0.01627 

8 

0.03657 

65 

—  0.01376 

0 

0.014(5(5 

9 

0.03563 

66 

—  0.01459 

+  1 

0.0130(5 

10 

0.03470 

67 

-  0.0154B 

2 

0.01147 

11 

0.03377 

68 

—  0.01626 

3 

0.00988 

12 

0.03284 

69 

—  0,01709 

4 

0.00830 

13 

0.03191 

70 

—  0.01792 

5 

0.00(572 

14 

0.03099 

71 

—  0.01874 

6 

0.00515 

15 

0.03007 

72 

—  0.01956 

7 

0.00359 

16 

0.02915 

73 

—  0.02038 

8 

0.00203 

17 

0.02822 

74 

—  0.02120 

9 

+  0.00047 

18 

0.02730 

75 

—  0.02202 

10 

—  0.00107 

19 

0.02639 

7(5 

—  0.02284 

11 

—  0.002(51 

20 

0.02548 

77 

—  0.023(56 

12 

—  0.0041.-. 

21 

0.02456 

78 

—  0.02447 

13 

—  0.00569 

22 

0.02364 

79 

—  0.02528 

14 

—  0.00721 

23 

0.02273 

80 

—  0.02(509 

15 

—  0.00873 

24 

0.02183 

81 

-  0.02(590 

16 

—  0.01'  •-'"> 

25 

0.02094 

82 

—  0.02771 

17 

—  0.0117(5 

26 

0.02004 

83 

—  0.02851 

18 

-0.01  21  i 

27 

0.01914 

84 

—  0.02932 

19 

—  0.0147(5 

28 

0.01824 

85 

—  0.03012 

20 

—  0.01(52(5 

29 

0.01734 

86 

—  0.03093 

21 

—  0.0177.r> 

30 

0.01645 

87 

—  0.03173 

22 

—  O.Oli  •'-':* 

31 

0.01555 

88 

—  0.03253 

23 

—  0.02071 

32 

0.01466 

89 

—  0.03333 

24 

—  0.0221'! 

33 

0.01377 

90 

—  0.03413 

25 

—  O.OL'3'6 

34 

0.01288 

91 

—  0.03492 

30 

—  0.03093 

+  35 

0.01200 

92 

—  0.03572 

+  35 

—  0.03*10 

APPENDICES 


257 


TABLE  I.    PULKOWA  REFRACTION  TABLES 

log  cr 


App't 

z 

••• 
log  <r 

App't 

z 

logo- 

o        / 

o         1 

80     0 

0.00019 

85    0 

0.00146 

80  30 

0.00022 

85  30 

0.00185 

81     0 

0.00025 

86    0 

0.00241 

81  30 

0.00029 

86  30 

0.00320 

82     0 

0.00035 

87    0 

0.00421 

82  30 

0.00045 

87  30 

0.00561 

83    0 

0.00057 

88    0 

0.00749 

83  30 

0.00073 

88  30 

0.01006 

84    0 

0.00091 

89    0 

0.01352 

84  30 

0.00116 

89  30 

0.01813 

85    0 

0.00146 

90    0 

0.02424 

Date 

i 

Jan.     15 

+  0.34 

Feb.    15 

+  0.27 

Mar.    15 

+  0.05 

April  15 

-  0.08 

May    15 

-  0.21) 

June    15 

-  0.26 

July    15 

-  0.33 

Aug.    15 

-  0.30 

Sept.   15 

-0.19 

Oct.     15 

+  0.16 

Nov.    15 

+  0.33 

Dec.    15 

.+  0.37 

log  tan  z  +  A  (log  B  +  log  T)  +  Xlog7  +  Hogo- 


TABLE  II.     PULKOWA  MEAN  REFRACTIONS 
Barom.  29.  5  inches,  Att.  TJierm.  50°  F.,  Ext.  Therm.  50°  F. 


App't 
z 

Mean 
refr'n 

App't 
z 

Mean 
refr'n 

App't 
z 

Meaii 
refr'n 

App't 

z 

Mean 
refr'n 

0   / 

/   n 

0     / 

i  it 

0     1 

/   ii 

0     / 

/  if 

0  0 

0  0.0 

58  0 

I  31.2 

73  0 

3  4.7 

80  40 

5  34 

5  0 

0  5.0 

59  0 

1  34.8 

73  20 

3  8.5 

81  0 

5  46 

10  0 

0  10.1 

60  0 

1  38.7 

73  40 

3  12.5 

81  20 

5  58 

15  0 

0  15.3 

61  0 

1  42.8 

74  0 

3  16.6 

81  40 

6  12 

20  0 

0  20.8 

62  0 

1  47.1 

74  20 

3  20.9 

82  0 

6  26 

25  0 

0  26.7 

63  0 

1  51.7 

74  40 

3  25.4 

82  20 

6  41 

30  0 

0  33.0 

64  0 

1  56.6 

75  0 

3  30.0 

82  40 

6  58 

32  0 

0  35.7 

65  0 

2  1.9 

75  20 

3  34.8 

83  0 

7  15 

34  0 

0  38.5 

65  30 

2  4.7 

75  40 

3  39.9 

83  20 

7  35 

36  0 

0  41.5 

66  0 

2  7.6 

76  0 

3  45.2 

83  40 

7  56 

38  0 

0  44.6 

66  30 

2  10.6 

76  20 

3  50.7 

84  0 

8  19 

40  0 

0  47.9 

67  0 

2  13.8 

76  40 

.3  56.5 

84  20 

8  43 

42  0 

0  51.4 

67  30 

2  17:  1 

77  0 

4  2.5 

84  40 

9  10 

44  0 

0  55.1 

68  0 

2  20.5 

77  20 

4  8.8 

85  0 

9  40 

46  0 

0  59.1 

68  30 

'2  24.1 

77  40 

4  15.5 

85  30 

10  32 

48  0 

1  3.4 

69  0 

2  27.8 

78  0* 

4  22.5 

86  0 

11  81 

50  0 

1  8.0 

69  30 

2  31.7 

78  20 

4  29.8 

86  30 

12  42 

52  0 

1  13.0 

70  0 

2  35.7 

78  40 

4  37.6 

87  0 

14  7 

53  0 

1  15.7 

70  30 

2  39.9 

79  0 

4  45.7 

87  30 

15  49 

54  0 

1  18.5 

71  0 

2  44.4 

79  20 

4  64.4 

88  0 

17  55 

55  0 

1  21.4 

71  30 

2  49.1 

79  40 

5  3.5 

88  30 

20  33 

56  0 

1  24.5 

72  0 

2  54.0 

80  0 

5  13.1 

89  0 

23  53 

57  0 

1  27.8 

72  30 

2  59.2 

80  20 

5  23.4 

89  30 

28  11 

58  0 

1  31.2 

73  0 

3  4.7 

80  40 

5  34.3 

90  0 

33  51 

258 


PRACTICAL   ASTRONOMY 


„ 

TABLE  III.     m  = 


- 
sin  1" 


-,  or  m  = 


sm  1" 


«,  or 

*o-* 

771 

t,OT 

to-t 

m 

t,  or 
to-t 

m 

t,  or 
t0-t 

m 

*,or 
to-t 

m 

m  8 

// 

m  s 

// 

m   s 

" 

ni   s 

" 

m   s 

" 

0  0 

0.00 

4  0 

31.42 

8  0 

125.65 

12  0 

282.68 

16  0 

502.5 

0  5 

0.01 

4  5 

32.74 

8  5 

128.28 

12  5 

286.62 

16  5 

507.7 

0  10 

0.05 

4  10 

34.09 

8  10 

130.94 

12  10 

290.58 

16  10 

513.0 

0  15 

0.12 

4  15 

35.46 

8  15 

133.63 

12  15 

294.58 

16  15 

518.3 

0  20 

0.22 

4  20 

36.87 

8  20 

136.34 

12  20 

298.60 

16  20 

523.6 

0  25 

0.34 

4  25 

38.30 

8  25 

139.'08 

12  25 

302.64 

16  25 

529.0 

0  30 

0.49 

4  30 

39.76 

8  30 

141.85 

12  30 

306.72 

16  30 

534.3 

0  35 

0.67 

.4  35 

41.25 

8  35 

144.64 

12  35 

310.82 

16  35 

539.7 

0  40 

0.87 

4  40 

42.76 

8  40 

147.46 

12  40 

314.95 

16  40 

545.2 

0  45 

1.10 

4  45 

44.30 

8  45 

150.31 

12  45 

319.10 

16  45 

550.6 

0  50 

1.36 

4  50 

45.87 

8  50 

153.19 

12  50 

323.29 

16  50 

556.1 

0  55 

1.65 

4  55 

47.46 

8  55 

156.09 

12  55 

327.50 

16  55 

561.6 

0 

1.96 

5  0 

49.09 

9  0 

159.02 

13  0 

331.74 

17  0 

567.2 

5 

2.31 

5  5 

50.73 

9  6 

161.98 

13  5 

336.00 

17  5 

572.8 

10 

2.67 

5  10 

52.41 

9  10 

164.97 

13  10 

340.30 

17  10 

578.4 

15 

3.07 

5  15 

54.11 

9  15 

167.97 

13  15 

344.62 

17  15 

584.0 

20 

3.49 

5  20 

55.84 

9  20 

171.02 

13  20 

348.97 

17  20 

58*9.6 

25 

3.94 

5  25 

57.60 

9  25 

174.08 

13  25 

353.34 

17  25 

595.3 

30 

4.42 

5  30 

59.40 

9  30 

177.18 

13  30 

357.74 

17  30 

601.0 

35 

4.92 

5  35 

61.20 

9  35 

180.30 

13  35 

362.17 

17  35 

606.8 

40 

5.45 

5  40 

63.05 

9  40 

183.46 

13  40 

366.64 

17  40 

612.5 

45 

6.01 

5  45 

64.91 

9  45 

186.63 

13  45 

371.11 

17  45 

618.3 

50 

6.60 

5  50 

66.81 

9  50 

189.83 

13  50 

375.12 

17  50 

624.1 

1  55 

7.21 

5  55 

68.73 

9  55 

193.06 

13  55 

380.17 

17  55 

630.0 

2  0 

7.85 

6  0 

70.68 

10  0 

196.32 

14  0 

384.74 

18  0 

635.9 

2  5 

8.52 

6  5 

72.66 

10  5 

199.60 

14  5 

389.32 

18  5 

641.7 

2  10 

9.22 

6  10 

74.66 

10  10 

202.92 

14  10 

393.94 

18  10 

647.7 

2  15 

9.94 

6  15 

76.69 

10  15 

206.26 

14  15 

398.58 

18  15 

653.6 

2  20 

10.69 

6  20 

78.75 

10  20 

209.62 

14  20 

403.26 

18  20 

659.6 

2  25 

11.47 

6  25 

80.84 

10  25 

213.02 

14  25 

407.96 

18  25 

665.6 

2  30 

12.27 

6  30 

82.95 

10  30 

216.44 

14  30 

412.68 

18  30 

671.6 

2  35 

13.10 

6  35 

85.09 

10  35 

219.88 

14  35 

417.44 

18  35 

677.7 

2  40 

13.96 

6  40 

87.26 

10  40 

223.36 

14  40 

422.23 

18  40 

683.8 

2  45 

14.85 

6  45 

89.45 

10  45 

226.86 

14  45 

427.04 

18  45 

689.9 

2  50 

15.76 

6  50 

91.68 

10  50 

230.39 

14  50 

431.87 

18  50 

696.0 

2  55 

16.70 

6  55 

93.92 

10  55 

233.95 

14  55 

436.73 

18  55 

702.2 

3  0 

17.67 

7  0 

96.20 

11  0 

237.54 

15  0 

441.63 

19  0 

708.4 

3  5 

18.67 

7  5 

98.50 

11  5 

241.14 

15  5 

446.55 

19  5 

714.6 

3  10 

19.69 

7  10 

100.84 

11  10 

244.79 

15  10 

451.50 

19  10 

720.9 

3  15 

20.74 

7  15 

103.20 

11  15 

248.45 

15  15 

456.47 

19  15 

727.2 

3  20 

21.82 

7  20 

105.58 

11  20 

252.15 

15  20 

461.47 

19  20 

733.5 

3  25 

22.92 

7  25 

107.99 

11  25 

255.87 

15  25 

466.50 

19  25 

739.8 

3  30 

24.05 

7  30 

110.44 

11  30 

259.62 

15  30 

471.55 

19  30 

746.2 

3  35 

25.21 

7  35 

112.90 

11  35 

263.39 

15  35 

476.64 

19  35 

752.6 

3  40 

26.40 

7  40 

115.40 

11  40 

267.20 

15  40 

481.74 

19  40 

759.0 

3  45 

27.61 

7  45 

117.92 

11  45 

271.02 

15  45 

486.88 

19  45 

765.4 

3  50 

28.85 

7  50 

120.47 

11  50 

274.88 

15  50 

492.05 

19  50 

771.9 

3  55 

30.12 

7  55 

123.05 

11  55 

278.76 

15  55 

497.23 

19  55 

778.4 

4  0 

31.42 

8  0 

125.65 

12  0 

282.68 

16  0 

502.46 

20  0 

784.9 

APPENDICES 


259 


TABLE  III.    w=  sinl,,  »  or  m 


sinl' 


t,  or 
to-t 

m 

t,  or 

to-t 

m 

m    s 

» 

m    s 

» 

20   0 

784.9 

24   0 

1129.9 

20   5 

791.4 

24   5 

1137.8 

20  10 

798.0 

24  10 

1145.6 

20  15 

804.6 

24  15 

1153.6 

20  20 

811.3 

24  20 

ll(h.5 

20  25 

817.9 

24  25 

1169.5 

20  30 

824.6 

24  30 

1177.5 

20  35 

831.2 

24  35 

1185.5 

20  40 

838.0 

24  40 

1193.5 

20  45 

844.7 

24  45 

1201.5 

20  50 

851.6 

24  50 

1209.6 

20  55 

858.4 

24  55 

1217.7 

21   0 

865.3 

25   0 

1225.9 

21   5 

872.1 

25   5 

1234.1 

21  10 

879.0 

25  10 

1242.3 

21  15 

886.0 

25  15 

1250.5 

21  20 

893.0 

25  20 

1258.8 

21  25 

900.0 

25  25 

1267.1 

21  30 

907.0 

25  30 

1275.4 

21  35 

914.0 

25  35 

1283.8 

21  40 

921.1 

25  40 

1292.2 

21  45 

928.2 

25  45 

1300.5 

21  50 

935.2 

25  50 

1309.0 

21  55 

942.3 

25  55 

1317.4 

22   0 

949.5 

26   0 

1325.9 

22   5 

956.7 

26   5 

1334.4 

22  10 

963.9 

26  10 

1342.9 

22  15 

971.2 

26  15 

1351.4 

22  20 

978.5 

26  20 

1360.1 

22  25 

»  985.8 

26  25 

1368.7 

22  30 

993.2 

26  30 

1377.3 

22  35 

1000.6 

26  35 

1385.9 

22  40 

1008.0 

26  40 

1394.7 

22  45 

1015.4 

26  45 

1403.4 

22  50 

1022.8 

26  50 

1412.2 

22  55 

1030.3 

26  55 

1420.9 

23   0 

1037.8 

27   0 

1429.7 

23   5 

1045.3 

27   5 

1438.5 

23  10 

1052.8 

27  10 

1447.4 

23  15 

1060.4 

27  15 

1456.3 

23  20 

1068.1 

27  20 

1465.2 

23  25 

1075.7 

27  25 

1474.1 

23  30 

1083.3 

27  30 

1483.1 

23  35 

1091.0 

27  35 

1492.1 

23  40 

1098.8 

27  40 

1501.1 

23  45 

1106.5 

27  45 

1510.2 

23  50 

1114.3 

27  50 

1519.2 

23  55 

1122.0 

27  55 

1528.3 

24   0 

1129.9 

28   0 

1537.5 

2  sin*** 

~  81 

nl" 

n 

m 

s 

" 

0 

0 

0.00 

2 

0 

0.00 

4 

0 

0.00 

6 

0 

0.01 

8 

0 

0.04 

9 

0 

0.06 

10 

0 

0.09 

11 

0 

0.14 

12 

0 

0.19 

12 

30 

0.23 

13 

0 

0.26 

13 

30 

0.31 

14 

0 

0.36 

14 

30 

0.41 

15 

0 

0.47 

15 

30 

0.54 

16 

0 

0.61 

16 

30 

0.69 

17 

0 

0.78 

17 

30 

0.88 

18 

0 

0.98 

18 

30 

1.09 

19 

0 

1.22 

19 

30 

1.35 

20 

0 

1.49 

20 

20 

1.60 

20 

40 

1.70 

21 

0 

1.82 

21 

20 

1.93 

21 

40 

2.06 

22 

0 

2.19 

22 

20 

2.32 

22 

40 

2.46 

23 

0 

2.61 

23 

20 

2.77 

23 

40 

2.93 

24 

0 

3.10 

24 

20 

3.27 

24 

40 

3.45 

25 

0 

3.64 

25 

20 

3.84 

25 

40 

4.05 

26 

0 

4.26 

26 

20 

4.48 

26 

40 

4.72 

27 

0 

4.96 

27 

20 

5.20 

27 

40 

5.46 

28 

0 

5.73 

ADDENDUM 

(SEE  PAGE  75,  LINES  2-5) 

The  effect  of  refraction  is  to  move  a  star's  apparent 
place  upward  on  its  vertical  circle;  and  as  vertical  circles 
converge  as  they  approach  the  zenith,  it  follows  that  the 
computed  distance  S'C  is  greater  than  the  observed  dis- 
tance. The  value  of  R  given  by  (60)  is  therefore  too 
great. 

Consider  the  triangle  whose  vertices  are  $',  (7,  and 
the  zenith  Z.  Neglecting  small  quantities  of  the  second 
order,  we  may  assume  the  angle  at  C  to  be  a  right  angle, 

and  we  have 

sin  CS'  =  sin  ZS'  sin  S'ZC. 


If  refraction  changes  the  zenith  distance  ZtS'by  — 
the  corresponding  change  in  CS'  is 

d(CS')  =  -  tan  CS'  cotZS'  d(ZS'). 

Replacing  ZS'  by  &  —  <f)  (without  sensible  error)  for  a  star 
observed  at  upper  culmination,  CS'  by  (m  —  m^)R,  and 
by  the  mean  refraction  r,  we  obtain 


m  -  w?0 
or,  with  sufficient  accuracy, 

dR  =  -  R  tan  1"  cot  (8  -  <£)  r. 

If  a  star  is  observed  at  lower  culmination,  S  must  be 
replaced  by  180°  -  8. 

In  the  problem  on  pp.  75-77, 

R  =  45".042,  180°  -  8  -  <£  =±  49°  00',  r  =  60", 

and  therefore 

dR  =  -  0".011. 

The  corrected  value  of  R  is  45".031.  This  value  should 
be  substituted  for  that  quoted  on  pp.  82,  133,  and  173. 
The  resulting  change,  amounting  to  but  one  part  in  four 
thousand,  is  insignificant  on  p.  82,  is  unimportant  on 
p.  133,  and  increases  the  value  of  <£  by  0'M3  on  p.  173. 

260 


INDEX 


(THE  NUMBERS  REFER  TO  PAGBS) 


Aberration:  general,  of  a  star  in  the 
direction  of  the  observer's  motion, 
40;  annual,  denned,  41;  annual, 
affecting  a  star's  apparent  place, 
45,  61;  annual,  in  right  ascension 
and  declination,  62,  64;  diurnal, 
denned,  41;  diurnal,  in  hour  angle 
and  declination,  41  ;  diurnal,  in 
azimuth,  43,  193,  200;  diurnal,  in 
azimuth  and  altitude,  43. 

Altitude:  defined,  3;  measured  at  sea 
with  a  sextant,  36,  92 ;  measured  on 
land  with  a  sextant,  93. 

Angle :  measured  by  vernier,  65 ;  meas- 
ured by  reading  microscope,  67,  179. 

Apparent  place :  defined,  25, 45 ;  reduc- 
tion to,  61. 

Axis  of  the  celestial  sphere,  defined,  3. 

Azimuth:  defined,  3;  constant  of  a 
transit  instrument,  127,  —  determina- 
tion of,  142;  of  a  point,  from  obser- 
vations of  a  star  near  elongation, 
193;  of  a  point,  from  observations 
of  Polaris  at  any  hour  angle,  199 ; 
sometimes  measured  from  north 
point,  199;  of  the  polar  axis  of  an 
equatorial  telescope,  217. 

Azimuth  and  altitude :  as  coordinates, 
7 ;  of  a  star,  from  given  hour  angle 
and  declination,  10. 

Barometer:  factor  in  refraction  for- 
mula?, 34,  35,  255. 

Berliner  Astronomisches  Jahrbuch,  2. 

Bessel:  his  star  numbers,  62;  his  star 
constants,  62. 

Celestial  sphere :  defined,  2. 
Chronograph:  described,  86;  illustra- 
tion   of,   87 ;    used    for    comparing 


clocks  and  chronometers,  85. 


Chronographic  method  of  recording 
transit  observations,  88. 

Chronometer:  correction,  83;  rate,  83; 
care  of,  85;  transported  for  deter- 
mining geographical  longitude,  159 ; 
determination  of  correction  from 
sextant  observations,  101,  —  from 
transit  ^observations,  127,  146,  — 
from  surveyor's  transit  observa- 
tions, 203. 

Circle:  vertical,  defined,  3;  hour,  de- 
nned, 3;  latitude,  defined,  5;  pri- 
mary, 7 ;  secondary  7 ;  graduated 
vertical,  175,  203;  graduated  hori- 
zontal, 191. 

Circummeridan  altitudes :  of  the  sun, 
for  determining  geographical  lati- 
tude, 112. 

Clock:  astronomical,  86 ;  driving,  212. 

Collimation:  axis,  122;  plane,  122; 
constant  of  a  transit  instrument, 
127,  — determination  of,  137,  151. 

Collimator:  described,  138;  determi- 
nation of  collimation  constant  by 
'one,  138,  —  by  two  collimators,  141; 
determination  of  flexure  by  two  col- 
limators, 181. 

Colure:  defined,  4. 

Connaissance  des  temps,  2. 

Coordinates :  spherical,  6 ;  systems  of, 
7 ;  transformation  of,  8. 

Day :  sidereal,  15 ;  true  solar,  16 ;  mean 
solar,  16 ;  civil,  17 ;  astronomical,  17. 

Declination:  defined,  4;  fundamental 
determination  of,  187 ;  differential 
determination  of,  187 ;  axis  of  equa- 
torial telescope,  212;  circle  of  equa- 
torial telescope,  212. 

Dip  of  the  horizon,  36. 


Distance  between  two  stars,  14. 
261 


262 


INDEX 


Earth:  form  and  dimensions  of,  23; 
radius  of,  25 ;  reduction  of  observa- 
tions to  the  center  of,  23. 

Eccentricity  of  a  graduated  circle,  09 ; 
of  a  sextant,  determined,  97. 

Ecliptic :  denned,  4 ;  obliquity  of,  4 ; 
true,  45 ;  mean,  45. 

Elongation:  of  a  star,  77,  193;  reduc- 
tion to,  194,  258. 

Ephemeris :  defined,  2 ;  American,  2. 

Equation  of  time :  defined,  17. 

Equator:  celestial,  defined,  3;  reading 
of  a  circle,  187. 

Equatorial  telescope:  described,  212; 
illustration  of,  213;  adjustments  of, 
214. 

Equinoxes:  defined,  4;  precession  of, 
44. 

Error  of  runs:  defined,  68;  determina- 
tion of  value  of,  68,  190. 

Eye  and  ear  method  of  observing,  85. 

Filar  micrometer:  described,  70;  illus- 
tration of,  71 ;  methods  of  using,  223, 
229,  233. 

Finder:  of  an  equatorial  telescope, 
212 ;  adjustment  of,  215. 

Geocentric  place :  defined,  25. 

Graduated  circle :  read  by  vernier,  65 ; 
read  by  microscope,  67 ;  eccentricity 
of,  69;  graduation  errors  of,  183; 
flexure  of,  184, 186. 

Hints  on  computing,  241. 

Horizon:  defined,  2;  dip  of,  36;  glass, 
89;  artificial,  93. 

Hour  angle :  defined ,  4 ;  of  a  star,  from 
given  zenith  distance  and  declina- 
tion, 12 ;  of  a  star,  from  given  right 
ascension  and  sidereal  time,  12. 

Hour  angle  and  declination :  as.  coor- 
dinates, 7;  of  a  star,  from  given 
azimuth  and  altitude,  8. 

Hour  circle:  defined,  3;  of  an  equa- 
torial telescope,  212. 

Index  correction  :  of  a  sextant,  defined, 
96,  —  determined  from  observations 
of  a  star,  9(5,  —  from  observations  of 
the  sun,  96. 

Interpolation:  elementary  considera- 
tions on,  19;  general  formula  for, 
244;  for  a  date  midway  between 
two  tabular  dates,  246. 


Latitude:  circles  of,  defined,  5;  of  a 
star,  5;  geographical,  5;  geocen- 
tric, 23;  reduction  to  geocentric, 
23;  geographical,  determined  from 
meridian  altitude  of  a  star  or  the 
sun,  109,  209,  —  from  an  altitude  of 
a  star,  110, —  from  circummeridian 
altitudes,  112,  — by  Talcott's  meth- 
od, 167,  —  from  meridian  circle 
observations,  187,  188,  190;  varia- 
tion of,  174. 

Least  reading  of  a  vernier,  (56. 

Least  squares:  application  to  deter- 
mination of  proper  motion,  58, — 
to  reduction  of  transit  observations, 
152,  —  to  combination  of  latitude 
observations,  173 ;  formulae  result- 
ing from,  247. 

Level:  spirit,  described,  79;  general 
formulae  for,  80;  adjustments  of, 
83;  value  of  a  division  of,  81 ;  strid- 
ing, 123;  zenith,  167;  plate,  196, 
211 ;  constant  of  transit  instrument, 
127,  — determination  of,  134,  140; 
trier,  81,  —  illustration  of,  81. 

Longitude :  of  a  star,  defined,  5 ;  geo- 
graphical, defined,  5;  geographical, 
determined  by  the  method  of  lunar 
distances,  115, — by  transportation 
of  chronometers,  159,  —  by  the  elec- 
tric telegraph,  160,  —  by  the  helio- 
trope, 163,  —  by  moon  culminations, 
164. 

Longitude  and  latitude :  as  coordi- 
nates, 7  ;  of  a  star,  from  given  right 
ascension  and  declination,  13. 

Lunar  distances :  method  of  determin- 
ing geographical  longitude  from, 
115. 

Meridian :  defined,  3 ;  direction  of, 
determined  from  observations  of  a 
star,  196,  200. 

Meridian  circle :  described,  175 ;  illus- 
tration of,  176 ;  flexure  of,  181 ;  deter- 
mination of  graduation  errors  of, 
183;  fundamental  and  differential 
determinations  of  declination,  187 : 
determinations  of  latitude,  187,  188, 
190. 

Meridian  mark,  or  mire,  143. 

Micrometer :  described,  70 ;  filar,  70,  — 
illustration  of,  71  ;  value  of  a  revolu- 
tion of,  72,  —  effect  of  temperature 
on  the,  77 ;  of  a  transit  instrument , 


INDEX 


263 


123 ;  of  a  zenith  telescope,  125,  167 ; 

of  a  meridian  circle,  179,  180 ;  ring, 

236. 
Microscope:  description  of  reading, 

67,  177;   methods  of  using,  67,  68, 

179, 190 ;  error  of   runs  of,  68,  179, 

190. 
Mire,  143. 

Nadir:  defined,  2;  determination  of 
the  level  and  collimation  constants 
of  a  transit  instrument  by  the 
method  of  the,  139,  140;  reading  of 
a  meridian  circle,  180. 

Nautical  Almanac/ American  Ephem- 
eris  and,  2. 

Nautical  Almanac  (British),  2. 

Noon:  sidereal,  15;  true  solar,  16;' 
mean  solar,  16. 

Nutation:  defined,  44;  affecting  the 
true  place  of  a  star,  45 ;  terms  in  re- 
duction formulae,  62,  64. 

Objects  for  the  telescope,  251. 

Obliquity  of  the  ecliptic :  defined,  4 ; 
affected  by  attractions  of  the  plan- 
ets, 47. 

Parallactic  angle :  defined,  11 ;  of  a 
star,  from  given  azimuth  and 
zenith  distance,  11 ;  of  a  star,  from 
given  hour  angle  and  declination, 
11. 

Parallax :  defined,  25 ;  equatorial  hori- 
zontal, 26 ;  of  a  body  in  zenith  dis- 
tance, the  earth  being  regarded  as  a 
sphere,  26;  of  a  body  in  azimuth 
and  zenith  distance,  the  earth  being 
regarded  as  a  spheroid,  27;  of  a  body 
in  right  ascension  and  declination, 
31 ;  factors,  32. 

Personal  equation  :  absolute,  157 ;  rela- 
tive, 157 ;  machine,  159. 

Plumb  line :  local  deviations  of,  23. 

Polar  axes :  of  an  equatorial  telescope, 
212;  adjustments  of,  215,  217. 

Polar  distance :  defined,  4. 

Poles  of  the  equator,  3. 

Precession :  luni-solar,  47  ;  planetary, 
47 ;  general,  47 ;  in  right  ascension 
and  declination  during  any  interval 
of  time,  48 ;  annual  value  of,  in  right 
ascension  and  declination,  50;  secu- 
lar variation  of  annual,  57. 

Prime  vertical :  defined,  3. 


Prismatic  sextant,  91. 

Prismatic  transit  instrument,  125. 

Proper  motion:  of  the  stars,  45,  54; 
determination  of  annual,  54 ;  reduced 
from  one  epoch  to  another,  55 ;  de- 
termination of,  by  the  method  of 
least  squares,  58. 

Reduction  to  elongation :  in  the  deter- 
mination of  azimuth,  194 ;  tables  for, 
258. 

Reduction  to  the  meridian :  of  circum- 
meridian  altitudes  of  a  star  or  the 
sun,  for  latitude,  112, 258 ;  for  zenith 
telescope  observations  for  latitude, 
171;  for  meridian  circle  determina- 
tions of  declination,  187. 

Refraction:  general  laws  of,  32;  ab- 
normal, in  azimuth,  33;  Pulkowa 
formula  for,  in  zenith  distance,  34; 
Pulkowa  tables  for  computing,  254 ; 
Pulkowa  mean,  257  ;  Comstock's 
formula  for,  35 ;  differential  formu- 
lae for,  in  right  ascension  and  decli- 
nation, 36;  differential,  affecting 
zenith  telescope  observations  for 
latitude,  169,  170;  correction  for,  in 
meridian  circle  observations,  187 ; 
differential,  affecting  filar  microm- 
eter observations,  224,  229,  234. 

Right  ascension :  defined,  5 ;  of  a  star, 
from  given  hour  angle  and  sidereal 
time,  12;  of  a  star,  from  transit  ob- 
servations, 177;  of  the  moon,  from 
transit  observations,  165. 

Right  ascension  and  declination:  as 
coordinates,  7. 

Ring  micrometer:  described,  236;  de- 
termination of  radius  of,  236 ;  method 
of  observing  with,  239. 

Semidiameter:  correction  for,  38,  92, 
93;  apparent,  of  the  moon,  38;  con- 
traction of,  produced  by  refraction, 
39. 

Sequence  and  degree  of  corrections, 
43. 

Sextant :  described,  89 ;  illustration  of, 
90;  general  principles  of,  90;  pris- 
matic, 91;  methods  of  observing  with, 
92;  adjustments  of,  94;  correction 
for  index  error  of,  96 ;  correction  for 
eccentricity  of,  97;  determinations 
of  time,  101 ;  determinations  of  lati- 
tude, 109 ;  observations  of  lunar  dis- 


264 


INDEX 


tances  for  determining  geographical 
longitude,  115. 

Sidereal  time:  defined,  5;  of  a  star, 
from  given  hour  angle  and  right  as- 
cension, 12.  - 

Solstices:  defined,  4. 

Spherometer :  Harkness's,  for  investi- 
gating form  of  pivots,  137. 

Stars:  catalogues  of,  46,  58,  59,  223; 
undetermined,  177;  standard,  178; 
alias  of,  251. 

Sun:  mean,  16;  right  ascension  of 
mean,  17;  shades,  to  protect  ob- 
server's eyes,  90,  207. 

Surveyor's  transit:  determination  of 
time  by,  203,  —  of  latitude  by,  209, 
—  of  azimuth  by,  211. 

Talcott's  method  of  determining  lati- 
tude, 167. 

Telescope:  sextant,  89;  zenith,  167; 
equatorial,  212;  magnifying  power 
of,  220;  field  of  view  of,  221. 

Thermometer:  attached,  refraction 
factor  for,  34,  255 ;  external,  refrac- 
tion factor  for,  34,  35,  256. 

Time :  sidereal,  15 ;  true  solar  or  ap- 
parent, 16 ;  mean  solar,  16;  civil,  17 ; 


astronomical,  17;  equation  of,  17; 
conversion  of,  18 ;  determined  f roua 
sextant  observations,  101,  —  from 
star  transits,  146,  —  from  surveyor's 
transit  observations,  203. 
Transit  instrument:  described,  122;. 
illustrations  of,  124,  126 ;  general 
formula?  for,  129;  wire  intervals  of, 
130,  132;  reduction  to  middle  wire 
of,  131;  determination  of  level  con- 
stant of,  134,  140,  —  of  inequality  of 
pivots  of,  134,  —  of  collimation  con- 
stant of,  137,  151,  —  of  azimuth  con- 
stant of,  142;  flexure  of  prismatic 
form  of,  157. 

Vernier:  described,  65;  illustration 
of,  66. 

Year:  tropical,  17;  the  fictitious,  46. 

Zenith:  defined,  2;  level,  167;  read- 
ing of  a  circle,  180. 

Zenith  distance:  defined,  3;  of  a  star, 
at  greatest  elongation,  78,  193. 

Zenith  telescope :  Talcott's  method  of 
determining  latitude  by,  167. 


DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

,  MATHEMATICS- 

This  book  is  due  on  the  last  elate  stamped  below,  or 

on  the  date  to  which  renewed. 
Renewed  books  are  subject  to  immediate  recall. 


Ifec'dUCB  A/M/S 


1985 


•nia 


UNIVERSITY  OF  CAUFORNIA  LIBRARY 


